Trigonometric real form of the spin RS model of Krichever and Zabrodin

We investigate the trigonometric real form of the spin Ruijsenaars-Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable 'free' system carried by a spin extension of the Heisenberg double of the ${\rm U}(n)$ Poisson-Lie group. The Poisson structure on the unreduced real phase space ${\rm GL}(n,\mathbb{C}) \times \mathbb{C}^{nd}$ is the direct product of that of the Heisenberg double and $d\geq 2$ copies of a ${\rm U}(n)$ covariant Poisson structure on $\mathbb{C}^n \simeq \mathbb{R}^{2n}$ found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity, and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars-Schneider system and we prove its degenerate integrability.

Two kinds of spin many-body models are studied in the literature. Those that feature only 'collective spin variables' belonging to some group theoretic phase space such as a coadjoint orbit, and those that have 'spin-vectors' embodying internal degrees of freedom of the interacting particles. The former type of models arise rather naturally in harmonic analysis and its classical mechanical counterpart [12,13,21,22,34,47]. The latter type of models, built on 'individual spins', were introduced at the non-relativistic level by Gibbons and Hermsen [24], and their 'relativistic' generalization was later put forward by Krichever and Zabrodin [32].
In fact, in 1995 Krichever and Zabrodin introduced a family of spin RS models at the level of equations of motion and posed the question of their Hamiltonian structure and integrability. These models have rational, trigonometric/hyperbolic and elliptic versions and are usually studied in the holomorphic category. The elliptic model encodes the dynamics of the poles of elliptic solutions of the 2D non-Abelian Toda lattice [32], and a special hyperbolic degeneration is related to affine Toda solitons [7]. The existence of a Hamiltonian structure was established by Krichever [31] in the general case based on a universal construction that is hard to make explicit (see also [53]). The rational case was treated via Hamiltonian reduction by Arutyunov and Frolov [4] in 1997, utilizing a 'spin extension' of the holomorphic cotangent bundle of GL(n, C). More than twenty years later, there appeared two different treatments of the holomorphic trigonometric/hyperbolic models: by Chalykh and Fairon [10] based on double brackets and quasi-Hamiltonian structures, and by Arutyunov and Olivucci [5] based on Hamiltonian reduction of a spin extension of the Heisenberg double [52] of the standard factorizable Poisson-Lie group structure on GL(n, C). In the present paper, we shall deal with the trigonometric real form of the models of [32] utilizing the Heisenberg double of the Poisson-Lie group U(n), which is a natural generalization of T * U(n). The spin extension of the Heisenberg double that we consider is based on a U(n) covariant Poisson structure on C n introduced by Zakrzewski [56].
Although the holomorphic systems are of great interest from several viewpoints, it is not easy to extract from them the features of the dynamics of the real forms, which should also be investigated. For motivation, it perhaps suffices to recall that all pioneering papers of the subject [8,38,50,54] are devoted to point particles moving along the real line or circle.
The C-valued dynamical variables of the Krichever-Zabrodin model are 'particle positions' x i (i = 1, . . . , n) together with d-component row vectors c i and column vectors a i . The composite spin variables F ij are built from these individual spins according to the rule 1) and the equations of motion can be written in first order form as follows: where x ik := x i − x k . In the elliptic case the 'potential' is given by V (x) = ζ(x) − ζ(x + γ) with the Weierstrass zeta-function and an arbitrary complex 'coupling constant' γ = 0. The model admits hyperbolic/trigonometric degenerations for which one has V hyp (x) = coth(x) − coth(x + γ) and V rat (x) = x −1 − (x + γ) −1 . The parameters λ i in (1.2) are arbitrary. This is a hallmark of gauge invariance, and thus it is natural to declare that the 'physical observables' are invariant with respect to arbitrary rescalings where the Λ i may depend on the dynamical variables as well. One way to deal with this ambiguity is to impose a gauge fixing condition. Note also the interesting feature of the model that the spins a i , c i are not purely internal degrees of freedom, since they directly encode the velocities through the equations of motionẋ i = F ii . In the trigonometric real form of our interest we put x j := 1 2 q j , where the q j are real and are regarded as angles. In other words, we deal with particles located on the unit circle at the points Q j := exp(iq j ). The spins c i and a i are complex conjugates of each other, and we parametrize them as where v(−) i is regarded as a d-component row vector. For each α, v(α) is also viewed as an n-component column vector, and thus F = α v(α)v(α) † is an n by n Hermitian matrix. The potential V is now chosen to be with a real, positive coupling constant γ. The gauge transformations are given by arbitrary Λ i ∈ U(1) and accordingly we have λ i ∈ iR. It can be checked that these reality constraints are consistent with the equations of motion (1.2). We remark in passing that they imply the second order equation . (1.6) The equations of motion as given above are local in the sense that one does not know on what phase space their flow is complete, which is required for an integrable system. Neglecting this issue, let us assume that α v(α) i = 0 for all i, which permits us to impose the gauge fixing condition Then, consistency with the requirement ℑ(U i ) = 0 can be used to uniquely determine the λ i , and one finds the gauge fixed equations of motion (1.9) In this paper, we develop a Hamiltonian reduction approach to the real, trigonometric spin RS model specified above. In particular, this yields a phase space on which all flows of interest are complete. An open dense subset of the phase space will be associated with the gauge fixing condition (1.7), and on this submanifold we shall determine the explicit form of the Poisson brackets that generate the equations of motion (1.8)

by means of the Hamiltonian
(1.10) We shall also prove the degenerate integrability of the model by displaying (2nd − n) independent, real-analytic integrals of motion that form a polynomial Poisson algebra whose ndimensional Poisson center contains H. These results will be derived by using the q i and gauge fixed versions of the 'dressed spins' v(α) as coordinates on the reduced phase space obtained from Hamiltonian reduction. However, we will put forward another remarkable set of variables as well, which consists of canonical pairs q i , p i and 'reduced primary spins' w α that decouple from q and p under the reduced Poisson bracket. On the overlap of their dense domains, the relation between the two sets of variables can be given explicitly, but the formula is very involved. The drawback of the variables q, p, w α is that in terms of them H and the equations of motion become complicated. Notice that the Newton equations (1.6) imply the conservation of the sum of the velocitieṡ q i , which gives the Hamiltonian via (1.8) and (1.10), andq i is non-negative by (1.8). The same features appear in the spinless chiral RS model [50] defined by the Hamiltonian (1.11) with Darboux coordinates 1 q i , θ j . The second order equations of motion for q i generated by this Hamiltonian reproduce the d = 1 special case of (1.6). To see this, note that F ij F ji = F ii F jj if d = 1, and substituteq i = 2F ii from (1.8) into (1.6). In fact, the spinless RS model results from the d = 1 special case of our Hamiltonian reduction: in this case w 1 becomes gauge equivalent to a constant vector and one derives the model utilizing also a canonical transformation between q, p mentioned above and q, θ [19]. Thus, the spin RS systems of [32] are generalizations of the chiral RS model. We follow the general practice in dropping 'chiral' from their name. Our result on the degenerate integrability of the system is not surprising, since the same property holds in the complex holomorphic case [5,10] and it also holds generically for large families of related spin many-body models obtained by Hamiltonian reduction [44,45,46]. Despite these earlier results, the degenerate integrability of our specific real system can not be obtained directly. Therefore, it requires a separate treatment, and we shall exhibit the desired integrals of motion in explicit form.
Here is an outline of this work and its main results. In Section 2, we present the master phase space M which is an extension of the Heisenberg double of U(n) by a space of primary spins. The latter space is formed of d ≥ 2 copies of C n endowed with a U(n) covariant Poisson bracket and a (Poisson-Lie) moment map, see Proposition 2.1. We also introduce the 'free' degenerate integrable system on M that will be reduced. In Section 3, we define the Hamiltonian reduction and progress towards the description of the corresponding reduced phase space M red , which is a real-analytic symplectic manifold of dimension 2nd. In particular, we exhibit two models of dense open subsets of M red ; the first one is used in the subsequent sections to derive the real form of the trigonometric spin RS system described above, while the second one allows us to prove that M red is connected and it leads to a concise formula for the reduced symplectic form (see Theorem 3.14 and Corollary 3.15). The second model will also be used for recovering the Gibbons-Hermsen system through a scaling limit (see Remark 3.16). In Section 4, we characterize the projection of a family of free Hamiltonian vector fields of M onto M red , and show in Corollary 4.3 that one of these projections reproduces the equations of motion (1.8). Then, in Section 5, we obtain the reduced Poisson bracket presented in Theorem 5.8. This offers an alternative way to derive the equations of motion (1.8), and we also provide a formula for the Poisson bracket of the Lax matrix that generates the commuting reduced Hamiltonians, see Proposition 5.10. In Section 6, we demonstrate the degenerate integrability of the real trigonometric spin RS system, with the final result formulated as Theorem 6.7. Section 7 concludes this work and gathers open questions. There are four appendices devoted to auxiliary results and proofs.
Note on conventions. The sign function sgn is such that sgn(i − k) is +1 if i > k, −1 if i < k, and 0 for i = k. Similarly to Kronecker's delta function, we define for any condition c the symbol δ c which equals +1 if c is satisfied, and 0 otherwise. For example, δ (j<l≤k) equals +1 if j < l and l ≤ k, while it is 0 if one of those two conditions is not satisfied.
2 Heisenberg double, primary spins and 'free' integrable system Eventually, we shall obtain the real, trigonometric spin RS system by reduction of an 'obviously integrable' system on the phase space M := M ×C n×d , where M is the Heisenberg double of the Poisson-Lie group U(n) and C n×d is the space of the so-called primary spin variables. In this section we present a quick overview of these structures, to be used in the subsequent sections. More details can be found in the references [16,19,29,30,35,52] and in Appendix A.

The Heisenberg double and its models
Let us start with the real vector space direct sum where b(n) denotes the Lie algebra of upper triangular complex matrices having real entries along the diagonal, and the unitary Lie algebra u(n) consists of the skew-Hermitian matrices. These are isotropic subalgebras with respect to the non-degenerate, invariant bilinear form of gl(n, C) given by X, Y := ℑtr(XY ), ∀X, Y ∈ gl(n, C), (2.2) which means that we have a Manin triple at hand. Then define using the projection operators with ranges u(n) and b(n), associated with the decomposition (2.1). For any X ∈ gl(n, C), we may write As a manifold, M is the real Lie group GL(n, C), and for any smooth real function f ∈ C ∞ (GL(n, C)) we introduce the gl(n, C)-valued derivatives ∇f and ∇ ′ f by where K denotes the variable running over GL(n, C). The commutative algebra of smooth real functions, C ∞ (M), carries two natural Poisson brackets provided by The minus bracket makes GL(n, C) into a real Poisson-Lie group, while the plus one corresponds to a symplectic structure on M. The former is called the Drinfeld double Poisson bracket and the latter the Heisenberg double Poisson bracket [52].
The real Poisson brackets can be extended to complex functions by requiring complex bilinearity. Then the real Poisson brackets can be recovered if we know all Poisson brackets between the matrix elements of K and its complex conjugate K. In the case of the Drinfeld double, we have and Consider the subgroup B(n) < GL(n, C) of upper triangular matrices having positive entries along the diagonal, and the unitary subgroup U(n) < GL(n, C). These subgroups correspond to the subalgebras in (2.1). It is well-known that both U(n) and B(n) are Poisson submanifolds of the Drinfeld double (GL(n, C), { , } − ). We denote their inherited Poisson structures by { , } U and { , } B , which makes them Poisson-Lie groups. The Poisson brackets on C ∞ (U(n)) and on C ∞ (B(n)) admit the following description. For any real function φ ∈ C ∞ (U(n)) introduce the b(n)-valued derivatives Dφ and D ′ φ by and for any χ ∈ C ∞ (B(n)) similarly introduce the u(n)-valued derivatives Dχ and D ′ χ. Then we have where the conjugation takes place inside GL(n, C). Similarly The opposite signs in the last two formulae are due to our conventions. By the Gram-Schmidt process, every element K ∈ GL(n, C) admits the unique decompositions and K can be recovered also from the pairs (g L , b L ) and (g R , b R ), by utilizing the identity These decompositions give rise to the maps Λ L , Λ R into B(n) and Ξ L , Ξ R into U(n), (2.14) Then we obtain the maps from GL(n, C) onto U(n) × B(n), which are all (real-analytic) diffeomorphisms. In particular we shall use the diffeomorphism to transfer the Heisenberg double Poisson bracket to C ∞ (U(n) × B(n)). The formula of the resulting Poisson bracket [16], called { , } 1 + , can be written as follows: for any F , H ∈ C ∞ (U(n) × B(n)). The derivatives on the right-hand side are taken at (g, b) ∈ U(n) × B(n); the subscripts 1 and 2 refer to derivatives with respect to the first and second arguments. As an application, one can determine the Poisson brackets between the matrix elements of (g, b) := (g R , b R ) on the Heisenberg double, which gives The same formulae are valid w.r.t. { , } 1 + , and this was used for the computation. Observe from the formula (2.17) that both Ξ R and Λ R are Poisson maps w.r.t. the Heisenberg double Poisson bracket and the Poisson brackets { , } U and { , } B , respectively. The same is true regarding the maps Ξ L and Λ L . A further property that we use later is that Note, incidentally, that Ξ L and Ξ R enjoy the analogous identity. The Heisenberg double admits another convenient model as well. This relies on the diffeomorphism between B(n) and the manifold P(n) = exp(iu(n)) of positive definite Hermitian matrices, defined by b → L := bb † . Then we have the diffeomorphism For ψ ∈ C ∞ (P(n)), define the u(n)-valued derivative dψ by dψ(L), X := d dt t=0 ψ(L + tX), ∀X ∈ iu(n). (2.22) This definition makes sense since (L + tX) ∈ P(n) for small t. By using m 2 , one can transfer the Poisson bracket { , } + to C ∞ (U(n) × P(n)). The resulting Poisson bracket is called { , } 2 + , and is given [16] by the following explicit formula: for any F, H ∈ C ∞ (U(n) × P(n)), where the derivatives with respect to the first and second arguments are taken at (g, L) ∈ U(n) × P(n). The subscript u(n) refers to the decomposition defined in (2.4). We end this review of the Heisenberg double by recalling the symplectic form, denoted Ω M , that corresponds to the non-degenerate Poisson structure { , } + . It can be displayed [1] as Here, dΛ L collects the exterior derivatives of the components of the matrix valued function Λ L . To be clear about our conventions, we remark that the wedge does not contain 1 2 , and the Hamiltonian vector field X h of h satisfies dh = Ω M ( , X h ) and {f, h} + = df (X h ) = Ω M (X h , X f ).

The primary spin variables
We begin by recalling that the real, trigonometric spin Sutherland model of Gibbons-Hermsen [24] type can be derived via Hamiltonian reduction of T * U(n) × C n×d , where C n×d ≃ R 2n×d carries its canonical Poisson structure. In particular, if the elements of C n×d are represented as a collection of C n column vectors w 1 , w 2 , . . . , w d , (2.25) then the d different copies pairwise Poisson commute. The symmetry group underlying the reduction is U(n), which acts on C n in the obvious manner, For our generalization, it is natural to require this to be a Poisson action, i.e., A (n) should be a Poisson map with respect to the Poisson structure (2.10) on U(n) and a suitable Poisson structure on C n . A further requirement is that the U(n)-action should be generated by a moment map. Specialized to U(n) with the Poisson structure (2.10), the notion of moment map that we use can be summarized as follows 2 . Suppose that we have a Poisson manifold (P, { , } P ) and a Poisson map Λ : P → B(n), where B(n) is endowed with the Poisson structure (2.11). Then, for any X ∈ u(n), the following formula defines a vector field X P on P: where the Poisson bracket is taken with every entry of the matrix Λ, and L X P denotes Lie derivative along X P . The map X → X P is automatically a Lie algebra anti-homomorphism, representing an infinitesimal left action of U(n). If it integrates to a global action of U(n), then the resulting action is Poisson, i.e., the action map A : U(n) × P → P is Poisson. In the situation just outlined, Λ is called the (Poisson-Lie) moment map of the corresponding Poisson action.
In the next proposition we collect the key properties of a Poisson structure on C n , which is a special case of the U(n) covariant Poisson structures found by Zakrzewski [56].
Proposition 2.1. The following formula defines a Poisson structure on C n ≃ R 2n : These formulae imply

30)
which means that the Poisson bracket of real functions is real. With respect to this Poisson bracket, the action (2.26) of U(n) with (2.10) is Poisson, and is generated by the moment map b : C n → B(n) given by

32)
The map b satisfies the identity The Poisson structure is non-degenerate, and the corresponding symplectic form is given by A variant of the factorization formula (2.33) (without connection to Zakrzewski's Poisson bracket) was found earlier by Klimčík, as presented in an unpublished initial version of [19]. For convenience, we give a self-contained proof of the proposition in Appendix A.
Definition 2.2. The pairwise Poisson commuting w 1 , . . . , w d with each copy subject to the Poisson brackets (2.28), (2.29) are called primary spin variables. The Poisson space obtained in this manner is denoted C n×d , { , } W , and we shall also use the notation W := (w 1 , . . . , w d ). (2.35) The corresponding symplectic form, Ω W , is the sum of d-copies of Ω C n (2.34), one for each variable w α , α = 1, . . . , d.

The unreduced 'free' integrable system
Let H be an Abelian Poisson subalgebra of the Poisson algebra of (smooth, real-analytic, etc.) functions on a symplectic manifold M of dimension 2N, such that all elements of H generate complete Hamiltonian flows. Assume that the functional dimension of H is r 3 , and that there exists also a Poisson subalgebra C of the functions on M whose functional dimensions is (2N −r) and its center contains H. Then H is a called an integrable system with Hamiltonians H and algebra of constants of motion C. Liouville integrability is the r = N, C = H, special case. One calls the system degenerate integrable (or non-commutative integrable, or superintegrable) if r < N. In the degenerate case, similarly to Liouville integrability, the flows of the Hamiltonians belonging to H are linear in suitable coordinate systems on the joint level surfaces of C. For further details of this notion and its variants, and for the generalization of the Liouville-Arnold theorem, we refer to the papers [27,37,39,45] and to Section 11.8 of the book [48]. is an Abelian Poisson subalgebra of C ∞ (M). We call the elements of H 'free Hamiltonians' since their flows are easily written down explicitly. Indeed, for H = Λ * R (h) the flow sends the initial value (K(0), W (0)) to (K(t), W (t)) = (K(0) exp (−tDh(b R (0))) , W (0)) . (2.41) It follows that b R and b L are constants along the flow and we have the 'free motion' on U(n) given by The functional dimension of H is n, and for independent generators one may take The invariance of these functions follows from the useful identity (2.44) The system is degenerate integrable, with C taken to be the algebra of all constants of motion, which are provided by arbitrary smooth functions depending on b L , b R and W . From the decomposition (2.12), we get 45) and this entails n relations between the functions of b L and b R . Thus the functional dimension of C is 2N − n, with N = n 2 + nd, as required. It is worth noting that the joint level surfaces of C are compact, since they can be viewed as closed subsets of U(n). There are several ways to enlarge H into an Abelian Poisson algebra of functional dimension N, i.e., to obtain Liouville integrability of the Hamiltonians in H. However, there is no canonical way to do so. Degenerate integrability is a stronger property than Liouville integrability, since it restricts the flows of the Hamiltonians to smaller level surfaces. For these reasons, we shall not pay attention to Liouville integrability in this paper.

Defining the reduction and solving the moment map constraint
We first describe a Poisson action of U(n) on M and use it for defining the reduction of the free integrable system. Then we shall deal with two parametrizations of the 'constraint surface', which is obtained by imposing the moment map constraint of equation (3.17) below.

Definition of the reduction
Let us start by introducing the following Poisson map Λ : M → B(n), and define the dressed spins v(α) and the half-dressed spins v α by the equalities The new variables on M given by are related by a diffeomorphism of U(n) × B(n) × C n×d to the variables Proof. We have the relations, which can be used to reconstruct the variables (3.5) from those in (3.4).
The statement analogous to Lemma 3.2 for the variables (g R , b R , v 1 , . . . , v d ) also holds. Later in the paper we shall use the following identities enjoyed by the half-dressed spins and the dressed spins, which are direct consequences of (2.33). These identities and the subsequent proposition actually motivated the introduction of these variables.
Proposition 3.4. The moment map Λ : M → B(n) given by (3.1) generates the action A : U(n) × M → M that operates as follows: (2.16). In other words, b L = Λ L (K) with K ∈ M parametrized by the pair (g R , b R ).
Proof. In order to avoid clumsy formulae and the introduction of further notations, in what follows we identify the variables g R , b R , B α , w α and so on with the associated evaluation functions on M, M and C n . We shall also use the infinitesimal dressing action corresponding to (2.38), which has the form For any X ∈ u(n), denote X M , X M , X C n the vector fields associated with the moment maps Λ : M → B(n), Λ L Λ R : M → B(n) and b : C n → B(n), respectively. The formula of X M is known [19,29] and X C n can be read off from §2.2. In fact, we have and By using these, application of the definition (2.27) to the real and imaginary parts of the evaluation functions gives From the last two equalities, we obtain In conclusion, we see that the vector field X M is encoded by the formula (3.13) together with which follow from (3.10) and the structure of Λ (3.1). The completion of the proof now requires checking that the formula (3.8) indeed gives a left-action of U(n) on M, whose infinitesimal version reproduces the vector field M found above. These last steps require some lines but are fully straightforward, and thus we omit further details.
Remark 3.5. The action (3.8) on M is called (extended) quasi-adjoint action, since if we forget the v(α) then it becomes the quasi-adjoint action on M that goes back to [29]. At any fixed (b R , g R ), the map η →η that appears in (3.8) is a diffeomorphism on U(n), and thus the quasi-adjoint action and the so-called obvious action have the same orbits. The obvious action, denoted A : U(n) × M → M, operates as follows: where v := (v(1), . . . , v(d)) and gv := (gv(1), . . . , gv(d)). The corresponding reduced phase space is According to Remark 3.5, it does not matter whether we use the quasi-adjoint or the obvious action for taking the quotient.
is naturally a Poisson algebra, with bracket denoted { , } red . This is obtained by using that the Poisson bracket of any two invariant functions is again invariant, and its restriction to Λ −1 (e γ 1 n ) depends only on the restrictions of the two functions themselves. One sees this relying on the first class [25] character of the constraints that appear in (3.17).
Since the elements of H (2.40) are U(n) invariant, they give rise to an Abelian Poisson subalgebra, H red , of C ∞ (M red ). The flows of the elements of H red on M red result by projection of the free flows (2.41), see Section 4.
Remark 3.6. By using that (3.17) can be written equivalently as ΛΛ † = e 2γ 1 n , it is not difficult to see that the triple (g R , b R , v) belongs to Λ −1 (e γ 1 n ) if and only if it satisfies We notice that the set of the triples (g R , b R b † R , v) subject to (3.19) is a subset of the set M × n,d,q defined in [10], which contains the elements (X, Z, 3.20) and the invertibility conditions where q is a nonzero complex constant, X, Z ∈ GL(n, C), A α ∈ C n×1 and B α ∈ C 1×n , for α = 1, . . . , d. (These are equations (4.3) and (4.4) in [10].) It is known that if q is not a root of unity, then the action of GL(n, C) on M × n,d,q , defined by is free. As explained in [9,10], this goes back to results in representation theory [11]. Direct comparison of (3.19) and (3.20), and of the corresponding group actions, shows that the U(n) action (3.15) on our 'constraint surface' Λ −1 (e γ 1 n ) is free. In our case the invertibility conditions (3.21) hold in consequence of the following identities that generalize (3.7): is a smooth symplectic manifold, whose Poisson algebra coincides with (C ∞ (M red ), { , } red ) presented in the preceding paragraph. Furthermore, since (M, Ω M ) is actually a real-analytic symplectic manifold and the formulae of the U(n) action and the moment map are all given by real-analytic functions, M red is also a real-analytic symplectic manifold. For the underlying general theory, the reader may consult [41], and also appendix D in [18].
Remark 3.7. We will eventually prove the degenerate integrability of the reduced system by taking advantage of the following functions on M: The identity (2.44) shows that L transforms by conjugation, and therefore these integrals of motion are invariant under the U(n) action (3.15) on M. Their real and imaginary parts descend to real-analytic functions on the reduced phase space.

Solution of the constraint in terms of Q and dressed spins
Our fundamental task is to describe the set of U(n) orbits in the 'constraint surface' Λ −1 (e γ 1 n ). For this purpose, it will be convenient to label the points of M by g R , L and v = (v(1), . . . , v(d)) using that L is given by (3.25). In the various arguments we shall also employ alternative variables.
Since g R can be diagonalized by conjugation, we see from the form of the U(n) action (3.8) (or (3.15)) that every U(n) orbit lying in the constraint surface intersects the set where T n is the subgroup of diagonal matrices in U(n). Below, stands for an element of T n , and Ad Q −1 denotes conjugation by Q −1 . For any γ ∈ R * , (e 2γ Ad Q −1 − id) is an invertible linear operator on gl(n, C), which preserves the subspace of Hermitian matrices. After this preparation, we present a useful characterization of M 0 .
, then L can be expressed in terms of Q and v as follows: For the matrix elements of L, this gives Conversely, if the Hermitian matrix L given by the formula (3.28) is positive definite, then where b R and the w α are viewed as functions of L and v, given by the invertible relations of equation (3.25) and Definition 3.1. We substitute this into the following equivalent form of the moment map constraint (3.17), and thus obtain the requirement By using Lemma 3.3 and the definitions of L and F , this in turn is equivalent to It follows that if (Q, L, v) ∈ M 0 , then L is given by the formula (3.28).
To deal with the converse statement, notice that L as given by the formula (3.28) is Hermitian and automatically satisfies (3.34), but its positive definiteness is a non-trivial condition on the pair (Q, v). Suppose that L (3.28) is positive definite. Then there exists a unique b R ∈ B(n) Then there exists unique w 1 , . . . , w d from C n for which (3.3) holds, and by (3.2) and (3.7) these variables satisfy Inserting into (3.35), and using (3.31), we see that (3.35) implies the constraint equation (3.32), whereby the proof is complete.
We have the following consequence of Proposition 3.8.
Corollary 3.9. Let L(Q, v) be given by (3.28) and define the set The formula (3.28) establishes a bijection between M 0 (3.26) and P 0 , which is an open subset of T n × C n×d .
Let us call Q regular if it belongs to where L(Q, v) is specified by (3.28). Notice that any g ∈ U(n) for which A g (3.15) maps an element of M reg 0 to M reg 0 must belong to the normalizer N (n) of T n inside U(n). Therefore, the quotient of the regular part of the constraint surface by U(n), denoted M reg red , can be identified as where the quotient refers to the restriction of the obvious U(n) action (3.15) to the subgroup In the next section, we present an alternative procedure for solving the moment map constraint (3.17), which will show that all elements of T n reg are admissible. . This is why we assumed that γ > 0. It would be desirable to describe the elements of the set P 0 (3.37) explicitly. For d = 1 the solution of this problem can be read off from [19]. On account of the next two observations, we expect that the structure of P 0 is very different for d < n and for d ≥ n. First, let us notice that Q = 1 n is not admissible if d < n, since in this case the rank of L given by the formula (3.28) is at most d, while the rank of any positive definite L is n. Second, note that if d ≥ n, then we can arrange to have F = 1 n by suitable choice of v. Let v 0 be such a choice. Then L(1 n , v 0 ) is a positive multiple of 1 n , and therefore there is an open neighbourhood of (1 n , v 0 ) in T n × C n×d that belongs to P 0 .

Solution of the constraint in terms of Q, p and primary spins
Now we return to using the variables g R , b R and W (2.35) for labeling the points of M. We can uniquely decompose every element b ∈ B(n) as the product of a diagonal matrix, b 0 , and an upper triangular matrix, b + , with unit diagonal. Applying this to and introduce also The moment map constraint on M 0 (3.26) reads Since b 0 drops out from the formula of Λ, it is left arbitrary, and we parametrize it as A crucial observation is that (3.43) can be separated according to the diagonal and strictly upper-triangular parts, since it is equivalent to the two requirements and The constraint (3.45) is responsible for a reduction of the primary spin variables. Next we make a little detour and present a general analysis of such reductions. Let us introduce the map φ : and notice from Remark A.5 that φ is the moment map for the ordinary Hamiltonian action of T n on the symplectic manifold (C n×d , Ω W ) of the primary spins. Here, the dual of the Lie algebra of the torus T n < U(n) is identified with the space b(n) 0 of real diagonal matrices. The torus action in question is given by Taking any moment map value from the range of φ, we define the reduced space of primary spins: Proposition 3.11. The moment map φ : i.e., the inverse image of any compact set is compact. Fixing any moment map value Γ for which γ j > 0 for all j, the reduced spin-space (3.50) is a smooth, compact and connected symplectic manifold of dimension 2n(d − 1).
Proof. We first prove that the map φ is proper. Since the compact sets of Euclidean spaces are the bounded and closed sets, and since φ is continuous, it is enough to show that the inverse image of any bounded subset of b(n) 0 ≃ R n is a bounded subset of C n×d . Due to the definition of φ and equation (2.31), the formula of φ = diag(φ 1 , . . . , φ n ) is determined by the equality The second equality shows that φ j (W ) ≥ 0. On the other hand, using the first equality and that G n+1 = 1, we see that if and only if Now, if diag(γ 1 , . . . , γ n ) is from a bounded set, then n k=1 γ k ≤ C with some constant C. By using this and the j = 1 special case of (3.53), we obtain (3.54) which implies that the inverse image of any bounded set is bounded. If γ j > 0 for all j, then we see from the formula (3.51) that for any W ∈ Φ −1 (Γ) and for each 1 ≤ j ≤ n there must exist an index 1 ≤ α(j) ≤ d such that This implies immediately that the action (3.48) of T n is free on φ −1 (Γ), and therefore C n×d red (Γ) (3.50) is a smooth symplectic manifold of dimension 2n(d − 1). Since the moment map φ is proper, all its fibers φ −1 (Γ) are compact, and by Theorem 4.1 in [26] they are also connected. Hence C n×d red (Γ) is also compact and connected.
Remark 3.12. If γ j > 0 for all j, then C n×d red (Γ) is actually a real-analytic symplectic manifold. To cover it with charts, for any map µ : {1, . . . , n} → {1, . . . , d} we introduce the set Then the reduced spin-space is the union of the open subsets Y (µ) := X(µ)/T n , and a model of One can specify coordinates on Z(µ) by solving the constraints (3.53) for the w µ(j) j in terms of the remaining free variables, the w α j with α = µ(j), which take their values in a certain open subset of C n(d−1) . It is an interesting exercise to fill out the details, and to also write down the reduced symplectic form by using these charts.
If d = 1, then the reduced spin-space consists of a single point. This is also true in the trivial case for which γ j = 0 for all j. If some of the γ j are zero and the others are positive, then the moment map constraint φ(W ) = Γ leads to a stratified symplectic space. Finally, note that for the case corresponding to equation (3.45) γ j = γ > 0 for all j. Now returning to our main problem, it is useful to recast (3.46) (3.58) By using the principal gradation of n × n matrices, this equation can be solved recursively for b + if S + (W ) and Q are given, with Q regular. In fact, the following lemma is obtained by a word-by-word application of the arguments of Section 5 in [15]; hence we omit the proof.
Lemma 3.13. Suppose that S + = S + (W ) and Q are given, with Q ∈ T n reg . Then equation (3.58) admits a unique solution for b + , denoted b + (Q, W ). Using the notation and for k = 2, . . . , n − a we have b a,a+k + = I a,a+k S a,a+k (3.61) Now we restrict ourselves to the regular part of M 0 , stressing that it is defined without reference to any particular parametrization: Any gauge transformation that maps an element of M reg 0 to M reg 0 is given by the obvious action (3.15) of the normalizer N (n) of T n inside U(n). The normalizer has the normal subgroup T n , and the corresponding factor group is the permutation group It is plain that M reg red is a dense, open subset of the reduced phase space, and the above consecutive quotients show that M reg 0 /T n is an S n covering space 4 of this dense open subset.
Theorem 3.14. By solving the moment map constraint for b R in the form b R = e p b + (Q, W ) as explained above, the manifold M reg 0 (3.62) can be identified with the model spacẽ where Γ = γ1 n . Utilizing this model, the covering space M reg 0 /T n of the regular part of the reduced phase space becomes identified with the symplectic manifold equipped with its natural product symplectic structure.
Proof. The restriction of the action (3.15) to T n translates into the action on the model spaceP reg 0 , from which we obtain the identification M reg 0 /T n ≃ T * T n reg × C n×d red (Γ) at the level of manifolds. Let ξ 1 :P reg 0 → M and ξ 2 : φ −1 (Γ) → C n×d denote the natural inclusions, and write Q j = e iq j . Then a simple calculation gives which proves the claimed identification at the level of symplectic manifolds.
Corollary 3. 15. The dense open submanifold M reg red ⊆ M red is connected, and consequently M red is also connected.
Proof. Since φ −1 (Γ) is connected by Proposition 3.11, the connected components ofP reg 0 correspond to the connected components of T n reg . It is well-known (see e.g. the appendix in [17]) that any two connected components of T n reg are related by permutations. Thus M reg red is the continuous image of a single connected component ofP reg 0 , implying its connectedness. The proof is finished by recalling that if a dense open subset of a topological space is connected, then the space itself is connected.
Remark 3. 16. In this long remark we explain how the (trigonometric real form of the) spin Sutherland model of Gibbons and Hermsen [24] can be obtained from our construction via a scaling limit. For this, we introduce a positive parameter ǫ and replace the variables p by ǫ p and W by ǫ By using this we see that the matrix b = b + in (3.41), given in explicit form by Lemma 3.13, has the expansion which is just the standard Hamiltonian of the (real, trigonometric) Gibbons-Hermsen model. Replacing γ by = ǫγ and taking the limit, the residual constraint (3.45) gives (w • j , w • j ) = 2γ. Then, rescaling not only the variables but also the symplectic form (3.68), one gets which reproduces the symplectic form of the Gibbons-Hermsen model. It is known [19] that the standard spinless RS Hamiltonian [50] can be derived as the reduction of tr(L) + tr(L −1 ) in the d = 1 case. For d ≥ 2, we shall show (see Corollary 4.3 and Corollary 5.9) that the Hamiltonian of the (real, trigonometric) spin RS model of Krichever of Zabrodin [32] is the reduction of tr(L). As was already discussed in the Introduction, the term chiral spin RS model could have been a more fitting name for the model of [32], but we follow the literature in dropping 'chiral' in this context.
In this subsection, we have derived an almost complete description of the reduced system. We have established that T * T n reg × C n×d red (Γ) is an S n covering space of a dense, open subset of the reduced phase space, and we can write down the Hamiltonians tr(L k ) by using the explicit formula b R = e p b + (Q, W ). Why is the paper not finished at this stage? Well, one reason is that although Q, p and W are very nice variables for presenting the reduced symplectic form, they are not the ones that feature in the the Krichever-Zabrodin equations of motion, which we wish to reproduce in our setting. In fact, the usage of the dressed spins v(α) (3.3) will turn out indispensable for this purpose. (Notationwise, we took this into account already in equation (1.8).) Another, closely related, reason is that the action of permutations is practically intractable in terms of the primary spins 5 . More precisely, the action on the components of Q is the obvious one, but on p and W it is known only in an implicit manner, via the realization of these variables as functions of L = b R b † R and the dressed spins, on which the full U(n) action, and thus also the permutation action, is governed by the simple formula (3.15). In short, both the formula b + (Q, W ) and the change of variables from Q, p, W to Q and dressed spins v(α) are complicated, and for some purposes the latter will prove to be more convenient variables.

The reduced equations of motion
Denoting the Hamiltonian vector field of H = Λ * R (h) by X H and viewing g R , v(α) and L as evaluation functions, in correspondence to (2.41), we have It is clear that X H admits a well-defined projection on M red , which encodes the reduced dynamics. Of course, one may add any infinitesimal gauge transformation to the vector field X H without modifying its projection on M red , i.e., instead of X H one may equally well consider any Y H of the form with arbitrary Z(g R , L, v) ∈ u(n). It is also clear that one may use the restriction of Y H to M 0 (3.26) for determining the projection, and Z can be chosen in such a manner to guarantee the tangency of the restricted vector field to M 0 .
Let us consider the vector space decomposition where u(n) 0 and u(n) ⊥ consist of diagonal and off-diagonal matrices, respectively. Accordingly, for any T ∈ u(n) we have More explicitly, setting Q = exp(iq), we have K kk = 0 and   . Now take an initial value (Q 0 , L 0 , v 0 ) ∈ M reg 0 and ǫ > 0 (ǫ = ∞ is allowed) such that g R (t) = exp(tV(L 0 ))Q 0 ∈ U(n) reg for − ǫ < t < ǫ, (4.11) where the elements of U(n) reg have n distinct eigenvalues. Notice from (2.42) that g R (t) describes the unreduced solution curve, and that a small enough ǫ will certainly do. Then, for −ǫ < t < ǫ there exists a unique smooth curve η(t) ∈ U(n) for which It is easy to see that (Q(t), L(t), v(α)(t)) given by the above Q(t) and yields the integral curve of the vector field (4.8) with Z = 0. We here used the property V(gLg −1 ) = gV(L)g −1 (∀g ∈ U(n), L ∈ P(n)), which follows from the definition (4.1). The auxiliary conditions imposed in (4.12) fix the ambiguity of the 'diagonalizer' η(t) of g R (t). The reduction approach leads to this solution algorithm naturally, but we should stress that an analogous algorithm was found long ago by Ragnisco and Suris [43] using a direct method. Then the evolution equation on M reg 0 corresponding to the vector field Y 0 H (4.8) with Z = 0 can be written explicitly as follows: it is not difficult to re-cast the off-diagonal matrix function K(Q, L(Q, v)) (4.7) in the form for all k = l. This gives (4.16) with (4.17). The validity of the last sentence of the corollary can also be checked directly. We shall confirm in §5.2 that the ensuing reduced Hamiltonian generates the equations of motion (1.8) via the reduced Poisson structure described in coordinates using the gauge fixing condition (1.7).

The reduced Poisson structure
The main purpose of this section is to present the explicit form of the reduced Poisson structure in terms of the variables that feature in the equations of motion (1.8). The first subsection contains a couple of auxiliary lemmae, in which we provide explicit formulae for the Poisson brackets of the half-dressed and dressed spins, and the matrix entries of g R and L. These permit us to establish that the U(n) invariant integrals of motion (3.24) form a closed polynomial Poisson algebra on the unreduced phase space, which automatically descends to the reduced phase space. This interesting algebra is given by Proposition 5.5. In the second subsection we utilize the Poisson brackets of another set of U(n) invariant functions in order to characterize the reduced Poisson structure. We shall rely on the fact that the restriction of the Poisson brackets of U(n) invariant functions to a gauge slice in the 'constraint surface' must coincide with the Poisson brackets of the restricted functions calculated from the reduced Poisson structure. All calculations required by this section are straightforward, but they are quite voluminous and not enlightening. We strive to give just enough details to provide the gist of these calculations, and so that an interested reader may reproduce them. Some of these details are relegated to Appendix B and Appendix C.

Some Poisson brackets before reduction
Using the results from Section 2, the Poisson structure { , } M on M can be described in terms of the (complex-valued) functions returning the entries of the matrices (g R , b R , w 1 , . . . , w d ) and their complex conjugates. Namely, we can use (2.7)-(2.8) with K = g R or K = b R , then (2.18)- (2.19) to characterize the Poisson structure restricted to functions on the Heisenberg double; for fixed α = 1, . . . , d, the Poisson brackets involving w α are given by (2.28)-(2.29). The Poisson brackets between functions of w α and functions of g R and b R vanish. Our aim is to translate these relations to the matrices (g R , , which are more convenient to understand the reduced phase space M red , see Section 3. As a first step, we express the Poisson structure on the half-dressed spins v α = b 1 · · · b α−1 w α defined in where 1 ≤ i, k ≤ n and 1 ≤ α, β ≤ d. In particular, this defines a Poisson structure on This result is proved in Appendix B. From the reality of the Poisson bracket, we have After appropriate rescaling, this reproduces the minus Poisson bracket introduced by Arutyunov and Olivucci in their treatment of the complex holomorphic spin RS system by Hamiltonian reduction [5]. Considering the analogous construction with the variables v α +,i := v d−α+1 i instead, we obtain the plus Poisson bracket introduced in [5]. From now on, we let b = b R , g = g R . Using Lemma 5.1 and the Poisson structure of the Heisenberg double, we can easily write the Poisson brackets involving the entries v(α) i of the dressed spins v(α) = b R v α .
The Poisson brackets of the dressed spins and the matrices b, g are given by the following formulae Now we present an interesting application of the above auxiliary results. Recall that our 'free Hamiltonians' (2.40) Poisson commute with the functions I k αβ defined in (3.24), and hence they Poisson commute with the elements of the polynomial algebra The algebra I L is finitely generated as a consequence of the Cayley-Hamilton theorem for L. We also note that for an arbitrary non-commutative polynomial P obtained as a linear combination of products of the matrices L and v(α)v(β) † , 1 ≤ α, β ≤ d, we have that tr(P) ∈ I L in view of the identity A key property of I L is that it is a Poisson subalgebra of C ∞ (M). This follows from the next result, which can be proved by direct calculation. which together with (5.18) implies that I L (5.16) is indeed a real Poisson algebra. Since the elements of I L are invariant with respect to the U(n) action on M, they descend to the reduced phase space. We shall further inspect these integrals of motion in Section 6.

The reduced Poisson bracket in local coordinates
In this subsection we shall derive explicit formulae for the reduced Poisson structure, restricting ourselves to an open dense subsetM reg red of the reduced phase space. More precisely, it will be more convenient to work on a covering space ofM reg red that supports residual S n gauge transformations.
We start by introducing the open dense subsetM reg 0 ⊂ M reg 0 (3.62), which is defined aš The corresponding open dense subset of the reduced phase space iš M reg red :=M reg 0 /N (n) .

(5.21)
Using that S n = N (n)/T n , we can take the quotient in two steps. Thus, similarly to (3.64), we haveM The main reason for introducing the particular gauge sliceM reg 0,+ for the free T n action oň M reg 0 is that S n still acts on it in the obvious manner, by permuting the n entries of Q and the components of each column vector v(α) ∈ C n . Similar 'democratic gauge fixing' was employed in the previous papers dealing with holomorphic systems [4,5,10]. The relation between the spaces just defined and those given in Section 3 is summarized in Figure 1. be the tautological inclusion. General principles of reduction theory [25,41] ensure that the pull-back ξ * Ω M is symplectic and satisfies ξ * Ω M =π * (Ω red ), whereπ :M reg 0,+ → M red is the canonical projection and Ω red is the reduced symplectic form. We let { , } red denote the Poisson bracket on C ∞ (M reg 0,+ ) that corresponds to ξ * Ω M (2.37), and note that it possesses the key property  In order to implement the above ideas, now we introduce the following U(n) invariant elements of C ∞ (M) Proof. The identities (5.28) are well-known. To establish (5.29) we use the decomposition (here then use for these three terms (5.9), (5.10) and (2.7) respectively. Some obvious cancellations yield (5.29). Finally, (5.30) only requires some of the Poisson brackets gathered in §5.1 and it can be proved in a way similar to (5.29).
Convenient variables onM reg 0,+ are provided by the evaluation functions Q j = e iq j ∈ U(1) and the real and imaginary parts of the v(α) j ∈ C. The latter are not all independent, since they obey the gauge fixing conditions It is clear that all these functions belong to C ∞ (M reg 0,+ ) and their mutual Poisson brackets completely determine { , } red .
The pull-backs of the functions (5.27) can be written in the local variables onM reg 0,+ as and we note that In conjunction with Lemma 5.7 and equation (5.26), these expressions can be used to determine the reduced Poisson brackets of the variables Q j and v(α). To state the result, we introduce the n × n matrix-valued functions S 0 and R α , 1 ≤ α ≤ d, whose entries are given by We also define the matrix S with entries S ij = S 0 ij − S 0 ij .
Theorem 5.8. In terms of the functions (Q j = e iq j , v(α) j ) defined onM reg 0,+ , and using the formulae (3.29) for L and (5.32) for U j , we can write the reduced Poisson bracket as The bracket {−, −} red is invariant under simultaneous permutations of the n components of the variables q and v(α) for α = 1, . . . , d.
The proof of this result is the subject of Appendix C. Let us already mention that the reader can check the reality condition We know from Corollary 4.3 that the projection of the Hamiltonian vector field of H = (e 2γ − 1) tr(L) onto the gauge sliceM reg 0,+ leads to the equations of motion (1.8). Of course, the corresponding reduced Hamiltonian must generate the same evolution equations via the reduced Poisson bracket. The reduced Hamiltonian is encoded by the pull-back H := ξ * H oň M reg 0,+ . Thus, the next result shows the consistency of the computations performed in Section 4 and Section 5.  Proof. We get from (5.37) thaṫ Noticing the identity where V (x) is the potential (1.5), this allows us to write Summing over k precisely givesv(α) i in (1.8) with (1.9).
As a second consequence of Theorem 5.8, we can write down the reduced Poisson brackets of the 'collective spins' (F ij ), which can be found in Appendix D. By using equation (3.29), then we can obtain the formula for the Poisson brackets of the entries of the Lax matrix oň M reg 0,+ , which implies that the symmetric functions of L are in involution. This is in agreement with the fact that H given in (2.40) is an Abelian Poisson subalgebra of C ∞ (M). To present the desired formula, we use the matrix S defined before Theorem 5.8. We also define and where E ab is the n × n elementary matrix with only nonzero entry equal to +1 in position (a, b).
Proposition 5.10. On the gauge sliceM reg 0,+ (5.23), the entries of the Lax matrix L (3.29) satisfy where t 12 = −s 12 + s 21 − r 12 . This relation implies that the functions tr(L k ) are in involution.
In (5.47), we used the standard notations L 1 = L ⊗ 1 n , L 2 = 1 n ⊗ L, and {L 1 , where the entries of L are seen as evaluation functions onM reg 0,+ .
Remark 5.11. The formulae of Theorem 5.8 exhibit an interesting two-body structure in the sense that the Poisson brackets of the basic variables with particle labels i and j close on this subset of the variables. This is consistent with the fact that the Hamiltonian (1.10) is the sum of one-body terms, while the equations of motion (1.8)-(1.9) reflect two-body interactions. It should be stressed that this interpretation is based on viewing q i and the dressed spin v(−) i as degrees of freedom belonging to particle i. The same features hold in the complex holomorphic spin RS models as well [4,5,10]. It is also worth noting that the formulae of Theorem 5.8 enjoy a nice homogeneity property. Namely, let us define a Z n -valued weight wt[−] by setting where e j ∈ Z n is +1 in its j-th entry and zero everywhere else. We can then observe from (5.37)-(5.39) that the reduced Poisson bracket preserves this weight.

Degenerate integrability of the reduced system
We discussed the degenerate integrability of the unreduced free system in §2.3, and now wish to show that this property is inherited by the reduced system. This is expected to hold not only in view of the earlier results on holomorphic spin RS systems [5,10] and related models [44,45,46], but also on account of a general result in reduction theory. In fact, it is known (Theorem 2.16 in [58], see also [27]) that the integrability of invariant Hamiltonians on a manifold descends generically to the reduced space of Poisson reduction. However, the pertinent spaces of group orbits are typically not smooth manifolds. The existing results provide strong motivation, but do not help us directly to establish integrability in our concrete case.
Our goal is to prove the degenerate integrability of the reduced system in the real-analytic category by explicitly displaying the required integrals of motion. Specifically, we wish to show that the n reduced Hamiltonians arising from the functions tr(L k ), k = 1, . . . , n, (6.1) are functionally independent, and that one can complement them to (2nd − n) independent functions using suitable reduced integrals of motion that arise from the real and imaginary parts of the U(n) invariant functions These integrals of motion appeared before in Proposition 5.5. As throughout the paper, we assume that d ≥ 2.
The independence of functions means linear independence of their exterior derivatives at generic points, and this can be translated into the non-vanishing of a suitable Jacobian determinant. For real-analytic functions, the determinant at issue is also real-analytic, and hence it is generically non-zero if it is non-zero at a single point. Thus, by patching together analytic charts, one sees that on a connected real-analytic manifold independence of real analytic functions follows from the linear independence of their derivatives at a single point. We can use this observation since we know (see Remark 3.6 and Corollary 3.15) that M red is a connected real-analytic manifold.

Construction of local coordinates
Our first goal below is to construct local coordinates around certain points of the reduced phase space in which the formulae of the integrals of motion become simple. The coordinates will involve the eigenvalues of L, whereby the Hamiltonians tr(L k ) acquire a trivial form. We start by noting that the moment map constraint admits solutions for which only a single one of the vectors v(α) is non-zero. Concerning those elements of Λ −1 (e γ 1 n ), the following useful result can be obtained from (the proof of) Lemma 5.2 of [19]. Lemma 6.1. Consider any y ∈ R n whose components y 1 , . . . , y n satisfy the inequalities y i > e 2γ y i+1 ∀i = 1, . . . , n with y n+1 := 0. (6.3) Then there exists (g 0 , L 0 , v 0 ) ∈ Λ −1 (e γ 1 n ) such that L 0 = diag(y 1 , . . . , y n ) and v(α) 0 = 0 for each 1 ≤ α < d (where d ≥ 2 and γ > 0). For such elements all components of the vector v(d) 0 are non-zero.
Proof. Given L 0 = diag(y 1 , . . . , y n ) and v(1) 0 = . . . = v(d − 1) 0 = 0, we have to find g 0 ∈ U(n) and v(d) 0 ∈ C n such that the moment map constraint (3.17) holds. Using (3.19), this means that This is equivalent to the requirement that there exists v(d) 0 ∈ C n such that L 0 + v(d) 0 (v(d) 0 ) † and e 2γ L 0 have the same spectrum. But this holds if and only if we have the equality of polynomials in λ (e 2γ y k − λ) . (6.5) We can expand the left-hand side as follows : Evaluating this identity at λ = y l yields which is positive due to (6.3). It now suffices to pick v(d) 0 whose components have moduli given by (6.8), while we pick for g 0 any unitary matrix diagonalizing Remark 6.2. Notice that a completely gauge fixed normal form of the elements appearing in Lemma 6.1 can be obtained by requiring all components of the vector v(d) 0 to be positive. We also note in passing that in the d = 1 case the set of possible (ordered) eigenvalues of L in (g R , L, v) ∈ Λ −1 (e γ 1 n ) is given [19] by the polyhedron in R n specified by the conditions y i ≥ e 2γ y i+1 for all i = 1, . . . , n − 1 and y n > 0.
Now we introduce two subsets of the inverse image of the 'constraint surface'.
The open subset S 1 ⊂ S is defined by imposing the further condition that the matrix is conjugate to diag(µ 1 , . . . , µ n ), where the µ i satisfy the inequalities e 2γ y i > µ i > e 2γ y i+1 ∀i = 1, . . . , n − 1 and e 2γ y n > µ n . (6.11) Note that S is non-empty since we can apply the analogue of Lemma 6.1 to obtain elements of Λ −1 (e γ 1 n ) for which only v(1) is non-zero, and S 1 is non-empty since for those elements L 1 = L. It is clear that S can serve as a model of an open dense subset of the reduced phase space. Below, we provide a full characterization of the elements of S 1 .
Taking y and µ subject to the inequalities in (6.11), define V(y, µ) ∈ R n by Observe that the function under the square root is positive; and the positive root is taken.
Furthermore, g R is of the form where g 0 R ∈ U(n) is a fixed solution of the constraint equation Conversely, take any positive definite L = diag(y 1 , . . . , y n ) and C n vectors v(1), . . . , v(d − 1) such that L and L 1 given by (6.10) satisfy the spectral conditions (6.11), and all components of v(1) are positive. Choose a diagonalizer g 1 according to (6.13) and define v(d) ∈ C n by the formula (6.14) using an arbitrary τ ∈ T n . Then equation (6.16) admits solutions for g R , the general solution has the form (6.15) with arbitrary Γ ∈ T n , and all so obtained triples (g R , L, v) belong to S 1 .
Proof. By using the definitions (3.25) of L and (6.10) of L 1 , we can always recast the moment map constraint (3.19) in the form (6.16), which implies the equality of characteristic polynomials By using (6.13) and introducingũ we can write the polynomial on the right-hand side of (6.17) as which is positive due to (6.11). We conclude from this and equation (6.19) that v(d) has the form (6.14). The claim (6.15) about the form of g R follows from (6.16) since L is diagonal and has distinct eigenvalues. The converse statement is proved by utilizing that the equality of the polynomials in λ (6.17) is equivalent to the existence of a unitary matrix g R that solves the constraint equation (6.16). Then we simply turn the above arguments backwards. The crux is that the spectral assumption (6.11) ensures the positivity of the expression in (6.12), whence v(d) can be constructed starting from the vectorũ = diag(τ 1 , . . . , τ n )V(y, µ).

From now on we write
with real-valued v(α) ℜ j , v(α) ℑ j . In the next statement we summarize how Lemma 6.4 gives us coordinates on S 1 . Corollary 6.5. Via the formulae of Lemma 6.4 for v(d) and g R , the elements of S 1 are uniquely parametrized by the 2n(d − 1) variables . . , n, α = 2, . . . , d − 1 (6.23) together with the 2n variables τ j ∈ U(1), Γ j ∈ U(1), j = 1, . . . , n. (6.24) The variables (6.23) take values in an open subset of R 2n(d−1) . The matrix elements of g 1 (6.13) can be chosen to be real-analytic functions of the 2n(d − 1) variables (6.23), and then the components of v(d) (6.14) are also real-analytic functions of these variables and the τ j . Likewise, the matrix elements of g 0 R can be chosen to be real-analytic functions of the variables (6.23) and the τ j . Consequently, the variables (6.23) together with t j and γ j in τ j = e it j and Γ j = e iγ j define a coordinate system on the open submanifold of the reduced phase space corresponding to S 1 .
Proof. The variables (6.23) run over an open set simply because the eigenvalues of L 1 depend continuously on them. This dependence is actually analytic since those eigenvalues are all distinct. Regarding the dependence of g 1 and g 0 R on the variables, we use the well-known fact that the eigenvectors of regular Hermitian matrices can be chosen as analytic functions of the independent parameters of the matrix elements.

Degenerate integrability
The reduced integrals of motion arising from (6.1) and (6.2) take a simple form in terms of our coordinates on S 1 . Relying on this, we shall inspect the following 2n(d − 1) reduced integrals of motion: where k = 1, . . . , n and α = 2, . . . , d − 1, and the additional 2n integrals of motion supplied by the real and imaginary parts of Proposition 6.6. The 2n(d−1) reduced integrals of motion (6.25), which include the n reduced Hamiltonians tr(L k ), are functionally independent on S 1 . On each connected component of S 1 , n further integrals of motion may be selected from the real and imaginary parts of the functions (6.26) in such a way that together with (6.25) they provide a set of 2nd−n independent functions.
Proof. We are going to prove functional independence by inspection of Jacobian determinants using the coordinates on S 1 exhibited in Corollary 6.5. Let us first consider the functions given by (6.25). If we order the 2n(d − 1) functions as written in (6.25) and also order the 2n(d − 1) coordinates as written in (6.23), then the corresponding Jacobian matrix J takes a block lower-triangular form, with n × n blocks. The first diagonal block, (∂trL k /∂y j ), is given by Y ∈ Mat n×n (R) with Y kj = ky k−1 j , while all other diagonal blocks are given by XD 1 with X kj = y k j and D 1 = diag(v(1) 1 , . . . , v(1) n ), except the second one, (∂I k 1,1 /∂v(1) j ), which equals 2XD 1 . By the definition of S 1 , the coordinates y j are positive and distinct while the v(1) j are positive, so that X, Y and D 1 are invertible. Hence J has rank 2n(d − 1).
To continue, consider the 2n functions ℜ(I k d,1 ), ℑ(I k d,1 ) with 1 ≤ k ≤ n. (6.27) It is clear that any function G taken from (6.25) satisfies ∂G/∂t j = 0. So our claim will follow if there exists a subset of n functions F 1 , . . . , F n from those in (6.27) for which the Jacobian matrix ∂F k ∂t l kl is invertible. Note from Lemma 6.4 and Corollary 6.5 that and therefore we can write the following equality of complex matrices where X is given by X kj = y k j as before. We have already established that all three factors in the above product of matrices are invertible. Thus ∂I d,1 /∂t is invertible, hence has rank n.
To finish the proof, it suffices to remark that the complex matrix ∂I d,1 /∂t is a complex linear combination of the rows of the matrix given in (6.30). If the latter matrix has rank strictly less than n, then so does ∂I d,1 /∂t, which gives a contradiction.
Let us recall from Proposition 5.5 that the unreduced phase space supports the polynomial Poisson algebra I L (5.16), whose Poisson center contains the polynomial algebra Since these Poisson algebras consist of U(n) invariant functions, they engender corresponding Poisson algebras I red L and H red tr over the reduced phase space M red . Our final result is a direct consequence of Proposition 6.6.
Theorem 6.7. The reduced polynomial algebras of functions H red tr and I red L inherited from H tr (6.33) and I L (5.16) have functional dimension n and 2nd − n, respectively. In particular, on the phase space M red of dimension 2nd, the Abelian Poisson algebra H red tr yields a real-analytic, degenerate integrable system with integrals of motion I red L .
Proof. Let us consider I red L and its Poisson center Z(I red L ). Denote r and r 0 the functional dimensions of these polynomial algebras of functions. Observe from Proposition 6.6 that r ≥ (2nd − n) and r 0 ≥ n. (6.34) The second inequality holds since H red tr is contained in Z(I red L ), and Proposition 6.6 implies that the functional dimension of H red tr is n. In a neighbourhood U 0 of a generic point of M red , we can choose a system of coordinates given by 2nd functions F 1 , . . . , F 2nd such that the first r functions belong to I red L , of which the first r 0 belong to Z(I red L ). In terms of such coordinates, the Poisson matrix P = ({F i , F j }) i,j can be decomposed into blocks as This matrix must be non-degenerate since the reduced phase space is a symplectic manifold.
In particular, this implies that the r 0 rows of B must be independent. Then the number of independent columns of B must be also r 0 , which cannot be bigger than the number of columns. This gives r 0 ≤ (2nd − r), or equivalently By combining (6.34) with (6.36), we obtain that r 0 = n and r = (2nd − n).
We see from the above proof that Z(I red L ) and H red tr have the same functional dimension. Since H red tr ⊆ Z(I red L ), we expect that these polynomial algebras of functions actually coincide.
Remark 6.8. Let us explain that our coordinates on S 1 are very close to action-angle variables.
To start, we recall that the joint level surfaces of the integrals of motion of the unreduced free system are compact, because (with the help of the variables (g R , L, W )) they can be identified with closed subsets of U(n). This compactness property is inherited by the reduced system. If we restrict ourselves to the open subset of the reduced phase space parametrized by S 1 , then the connected components of the joint level surfaces of the elements of I red L (5.16) are the n-dimensional 'Γ-tori' obtained by fixing all variables in (6.23) and (6.24) except the Γ j . Both the gauge slice S (6.9) and its subset S 1 are invariant under the flow (2.42) of the Hamiltonian H k := 1 2k tr(L k ), for every k = 1, . . . , n, which gives the following linear flow on the Γ-torus: where Γ 0 j refers to the initial value. This statement holds since for H k (L) ≡ h k (b R ) one has Dh k (b R ) = iL k . The flow (6.37) entails that on S 1 the variablesp j := 1 2 log y j are canonical conjugates to the angles γ j in Γ j = e iγ j , i.e., they satisfy {γ j ,p l } red = δ kl .

Conclusion
In this paper we investigated a trigonometric real form of the spin RS system (1.2) introduced originally by Krichever and Zabrodin [32] and studied subsequently in [5,10] in the complex holomorphic setting. We have shown that this real form arises from Hamiltonian reduction of a free system on a spin extended Heisenberg double of the U(n) Poisson-Lie group, and exploited the reduction approach for obtaining a detailed characterization of its main features. In particular, we presented two models of dense open subsets of the reduced phase space where the system lives. The model developed in §3.3 led to an elegant description of the reduced symplectic form (Theorem 3.14), while the equations of motion and the corresponding Hamiltonian are complicated in the pertinent variables based on the 'primary spins'. On the other hand, the model studied in §3.2 and in Sections 4 and 5 allowed us to recover the spin RS equations of motion (1.8) from the projection of a free flow (Corollary 4.3), but the reduced Poisson brackets (Theorem 5.8) take a relatively complicated form in the underlying 'dressed spin' variables. In our framework the solvability of the evolution equations by linear algebraic manipulations emerges naturally (Remark 4.2), and we also proved their degenerate integrability by explicitly exhibiting the required number of constants of motion (Theorem 6.7).
A basic ingredient of the unreduced phase space that we started with was a U(n) covariant Poisson structure on C n ≃ R 2n that goes back to Zakrzewski [56], for which we found the corresponding moment map (Proposition A.3) and symplectic form (Proposition A.6).
We finish by highlighting a few open problems related to our current research. As always in the reduction treatment of an integrable Hamiltonian system, one should gain as complete an understanding of the global structure of the reduced phase space as possible. The basic point is that the projections of free flows are automatically complete, but only on the full reduced phase space. In the present case, one should actually construct two global models of the reduced phase space: one fitted to the system that we have studied, and another one that should be associated with its action-angle dual. Without going into details, we refer to the literature [20,23,44,49] where it is explained that the integrable many-body systems usually come in dual pairs, and the same holds for their several spin extensions. In our case, the commuting Hamiltonians of the dual system are expected to arise from the reduction of the Abelian Poisson algebraĤ = Ξ * R (C ∞ (U(n))), which is in some sense dual to H (2.40) on which our system was built.
It could be interesting to explore generalizations of the construction employed in our study. For example, one may obtain new variants of the trigonometric spin RS model by replacing some or all of the primary spins w α by z α subject to the Poisson bracket described at the end of Appendix A (Remark A.7). Generalization of our reduction in which the Heisenberg double is replaced by a quasi-Hamiltonian double of the form U(n) × U(n) [2], and the primary spins are also modified suitably, should lead to compactified spin RS systems. It should be possible to uncover a reduction picture behind the hyperbolic real form of the spinless and spin RS models, too. All these issues, as well as the questions of quantization and the reduction approach to elliptic spin RS models, pose challenging problems for future work.

A Properties of the primary spin variables
In this appendix we first elaborate the properties of the primary spin variables that were summarized in Proposition 2.1. As was already mentioned, the pertinent Poisson structure on C n ≃ R 2n is a special case of the U(n) covariant Poisson structures due to Zakrzewski [56]. Nevertheless, to make our text self-contained, we shall also verify its Jacobi identity and covariance property. Then we present the corresponding moment map and symplectic form, which have not been considered in previous work.
For any real function F ∈ C ∞ (C n ), we define its C n -valued 'gradient' ∇F by the equality 6 where the elements of C n are viewed as column vectors. We note that any real linear function on the real vector space C n is of the form for some ξ ∈ C n , and for such function ∇F ξ = ξ. Next we give a convenient presentation of Zakrzewski's Poisson bracket.
Proposition A.1. For real functions F, H ∈ C ∞ (C n ), let ξ(w) := ∇F (w) and η(w) := ∇H(w). Then the following formula where the notation (2.4) is used, defines a Poisson bracket on C ∞ (C n ). Equivalently to the formula (A.3), the Hamiltonian vector field V H associated with H ∈ C ∞ (C n ) is given by Extending the real Poisson bracket to complex functions by complex bilinearity, the Poisson brackets of the component functions w → w i satisfy the explicit formulae (2.28) and (2.29).
Proof. The antisymmetry of the last two terms of (A.3) is obvious, while the antisymmetry of the sum of the first and second terms is seen from the identity where we used constant ξ and η for simplicity. Here and below, the subscripts u and b stand for u(n) and b(n).
Regarding the Jacobi identity, it is enough to verify it for linear functions F ξ , F η and F ζ for arbitrary ξ, η, ζ ∈ C n . In this verification we may use the formula (A.4), since this expresses the identity {F, H}(w) = ℑ(ξ(w) † V H (w)), and does not rely on the Jacobi identity.
We start by calculating the gradient of {F ξ , F η } from (A.3), and find By spelling this out, we obtain After making several self-evident cancellations, and using cyclic permutations to reorganize terms in a convenient way, we get It is not difficult to see that the first line gives zero. Rearranging the second line, we have Having verified the Jacobi identity, it remains to calculate the Poisson brackets of the components of w and their complex conjugates. Let e k (k = 1, . . . , n) denote the canonical basis of C n . One obtains by tedious calculation that the Hamiltonian vector fields of the linear functions given by the real and imaginary parts of the components w k have the following form: iℜ(w n )w n e n + ie n − 1 2 i(w n − w n )w k = n. and iℑ(w n )w n e n − e n − 1 2 (w n + w n )w k = n.
By using these, one can check that the formulae (2.28) and (2.29) follow. If desired, the reader can supply the details.
The bracket (A.3) has the nice property that the natural action of U(n) on C n is Poisson [56], and this can also be checked using linear functions F ξ . To this end, for any g ∈ U(n) and w ∈ C n we define the functions F ξ (g · ) ∈ C ∞ (C n ) and F ξ ( · w) ∈ C ∞ (U(n)) by F ξ (g · )(w) = F ξ (gw) = F ξ ( · w)(g). (A.8) Then an easy calculation gives that which in turn is equal to the value at g of the Poisson bracket (2.10) of the functions F ξ ( · w) and F η ( · w) on U(n). The last equality follows using DF ξ ( · w)(g) = (gwξ † ) b(n) and elementary manipulations. Thus, we have which means that the map U(n) × C n ∋ (g, w) → gw ∈ C n is indeed a Poisson map. Let us recall the diffeomorphism b → bb † (A.12) from the group B(n) to the space P(n) of positive definite Hermitian matrices. By this diffeomorphism, the Poisson structure (2.11) on B(n) is converted into a Poisson structure on P(n), which is given by the first term of (2.23), i.e. from C n to P(n) is Poisson.
Proof. Let X, Y ∈ u(n) and consider the pull-backs Φ * (f X ) and Φ * (f Y ) of the functions f X (L) := X, L and f Y (L) := Y, L . We have Using the formula (A.3) with (∇Φ * (f X ))(w) = −2Xw and similar for f Y , we can compute Here, we have taken into account that, for example, ℑtr(XY ) = 0 for X, Y ∈ u(n). The statement follows since the linear functions of the form f X can serve as coordinates on P(n).  17) with (A.14) is the moment map for the Poisson action (2.26) of U(n) on C n . According to (2.27), this means that we have ℑ (∇F (w)) † Xw = ℑtr X{F, b}(w)b(w) −1 , ∀X ∈ u(n), w ∈ C n , F ∈ C ∞ (C n ). (A. 18) Proof. For ease of notation, we verify the relation for linear functions F ξ on C n , which is sufficient. For this, we have to calculate the b(n)-valued function Since (A.12) is a diffeomorphism, β F is uniquely determined by 20) and this can be calculated as follows. First, we rearrange the expression (A.4) of the Hamiltonian vector field in the form Then, as (ξ † w − w † ξ) ∈ iR, we obtain (A.22) But the last two terms cancel, and hence we see that By using this, the right-hand-side of (A.18) becomes − ℑtr(Xβ F (w)) = ℑtr(Xwξ † ) = ℑ(ξ † Xw), (A. 24) whereby the proof is complete.
Remark A.4. We had no need for the explicit formula of b(w) in the above, but in some other calculations it is needed. The reader can verify directly that it obeys equation (2.31).
Remark A.5. The maximal torus T n < U(n) is a Poisson subgroup with vanishing Poisson bracket, and therefore the restriction of the U(n) action to T n gives an ordinary Hamiltonian action. One can identify the dual Poisson-Lie group of T n with B(n) 0 , the group of positive diagonal matrices, with zero Poisson bracket. Then the corresponding group valued moment map is provided by w → b(w) 0 , which is the diagonal part of b(w). Writing b(w) 0 = exp(φ(w)), (A. 25) we get the ordinary moment map w → φ(w) ∈ b(n) 0 , where b(n) 0 (the space of real diagonal matrices) is identified with the linear dual of u(n) 0 .
The following proposition represents one of the side results of the paper.
Proposition A.6. The Poisson bracket (A.3) is symplectic and, with G j = 1 + n k=j |w j | 2 , the corresponding symplectic form on C n is given by Proof. We start from the coordinate form of the Poisson bracket, copied here for convenience: We shall first invert the Poisson tensor on the dense open submanifold on which all |w j | > 0, where we can use the parametrization w j = e iϕ j |w j |. Let us consider sgn(r − k)|w r | 2 , (A.28) from which we easily obtain {|w i | 2 , |w k | 2 } = 0. (A.29) Using this, and restricting now to our submanifold, the relation (A.28) implies Plainly, we have the identity {w j , w k } + e 2iϕ j e 2iϕ k {w j , w k } = 2|w j w k |{e iϕ j , e iϕ k }. It is convenient to change variables, noting that the map (|w 1 | 2 , . . . , |w n | 2 ) → (G 1 , . . . , G n ) is invertible. Then, it is elementary to derive from (A.30) the relation that can be also written as 34) This means that the matrix of Poisson brackets, in the variables ϕ i , ln G k has the form with A = 1 n + B + B 2 + · · · + B n−1 , (A. 36) where B is the nilpotent matrix having the entries B ik = δ i,k+1 . Both A and P are invertible, and their inverses are If we now substitute the identities and dϕ k = (2i|w k | 2 ) −1 (w k dw k − w k dw k ), (A. 40) then Ω (A.38) takes the form The Poisson map b − can be used to introduce variants of our reduction. Concretely, one may replace one or more of the b factors in (3.1) by b − , and study the reduced system. The restriction γ > 0 in the moment map constraint (3.17) then might not be necessary. Let us also note that one obtains a Poisson pencil on C n if one replaces the last term of (A.3) by −λℑ ξ(w) † η(w) for any real parameter λ, and the formula (A.46) corresponds to λ = −1.

B Proof of Lemma 5.1
In this section, we work over (C n×d , { , } W ) with the primary spins (w α ), see §2.2. We set { , } := { , } W to simplify notations. As noted in §3.1, the half-dressed spins v α can be defined in C n×d in terms of the primary spins. It is convenient to introduce the matrices b α = b(w α ) and B α = b 1 · · · b α , so that v α = B α−1 w α .
Proof. From (5.28) and (5.33) we get for any M, N ∈ N, e iM q i e iN q j {q i , q j } red .
Considering this equality for M, N = 1, . . . , n, this is equivalent to whereÛ (0) ∈ Mat n×n (C) is given byÛ Proof. From (5.29), after summing over all α, β we get for any M, N ∈ N i,j Using Lemma C.2, we obtain i,j e iM q i U i e iN q j {U i , q j } red = − i U 2 i e i(M +N )q i .
For the second identity, we use (5.29) with summation over all β, and we get for any M, N ∈ N i,j Now that the first identity is proved, we can use it to get As before, we write this for N, M = 1, . . . , n as where the n × n matrices are given byÛ (2) kl = {v(α) k , q l } red , U (2) kl = −δ kl v(α) k . Again by invertibility of E andẼ, we getÛ (2) = U (2) .
The last identity follows from the second one by complex conjugation.
From now on, we do not provide complete proofs of the different results that are stated. They can be successively obtained by direct computations in the same way as we got Lemmae C.2 and C.3.

D Poisson brackets of collective spins
Recall the matrix (S ij ) defined before Theorem 5.8. The reduced Poisson brackets of the socalled collective spins F (3.29) can be computed in the following form.
Lemma D.1. Denoting q ab := q a − q b , the following identity holds onM reg This follows from Theorem 5.8 by direct calculation. The reader can easily check the reality condition {F ji , F lk } red = {F ij , F kl } red = {F ij , F kl } red . Taking i = j and k = l in Lemma D.1, everything cancels out except for the third line, which can be rewritten as follows: {F jj , F kk } red = F jk F kj 2 cot( q jk 2 ) 1 + sinh −2 (γ) sin 2 ( , for j = k . (D.1) Let us now assume that d = 1, so that F jk F kj = F jj F kk . Note that the formula of L (3.29) shows that F jj > 0. Motivated by the form of the equations of motion (1.8) and the spinless Hamiltonian (1.11), we make the change of variables F jj = e 2θ j i =j 1 + sinh 2 γ 1 + sin 2 q i −q j 2 1 2 . (D.2) Using (5.37) and (D.1), it turns out that (q j , θ j ) are Darboux variables, and we recover the standard chiral RS Hamiltonian (1.11) for H = j F jj .