A family of semitoric systems with four focus-focus singularities and two double pinched tori

We construct a 1-parameter family $F_t=(J, H_t)_{0 \leq t \leq 1}$ of integrable systems on a compact $4$-dimensional symplectic manifold $(M, \omega)$ that changes smoothly from a toric system $F_0$ with eight elliptic-elliptic singular points via toric type systems to a semitoric system $F_t$ for $ t^-<t<t^+$. These semitoric systems $F_t$ have precisely four elliptic-elliptic and four focus-focus singular points. Moreover, at $t= \frac{1}{2}$, the system has precisely two focus-focus fibres each of which contains exactly two focus-focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.


Introduction
Integrable systems lie at the intersection of many areas in mathematics and physics like, for example, dynamical systems, ODEs, PDEs, symplectic geometry, Lie theory, classical mechanics, mathematical physics etc. Integrable systems display a 'certain amount of order' due to being 'not chaotic' and having their flow lines stay in the fibres of the momentum map. Their global behaviour, nevertheless, can be very intricate. When it comes to their singularities, nondegenerate singularities can be classified by means of a local normal form that splits a singularity into hyperbolic, elliptic, regular, and focus components where in fact the latter always comes as a pair, usually thus referred to as focus-focus block.
In this paper, we will construct a special 1-parameter family of integrable systems of two degrees of freedom on a 4-dimensional manifold which displays an interesting bifurcation behaviour and where all mentioned types of components will appear except for the hyperbolic ones. Since a completely integrable system gives rise to a Lagrangian fibration this explicit family may also become of interest for the study of low dimensional singular Lagrangian fibrations and, for instance, the (non)displaceability of its fibers under Hamiltonian isotopies, i.e., questions of symplectic rigidity.
Let us now be more precise. Given a 4-dimensional, connected, symplectic manifold (M, ω), a semitoric system is a completely integrable system F := (J, H) : M → R 2 where F only has nondegenerate singularities that have no hyperbolic components and where J : M → R is proper and induces an effective Hamiltonian S 1 -action. For instance, coupled spin oscillators and coupled angular momenta are semitoric.
Contrary to toric systems on 4-dimensional manifolds that give rise to an S 1 × S 1 -action and only admit elliptic-elliptic and elliptic-regular singular points, semitoric systems give rise to an S 1 × R-action and admit in addition focus-focus singularities.
Whereas toric systems are classified (up to equivariant symplectimorphism) by precisely one invariant, namely by the image of their momentum map (cf. Delzant [De]), the classification of semitoric systems is more involved: apart from one invariant based on a 'straightened' image of the momentum map, Pelayo & Vũ Ngo . c [PV1,PV2] showed that there are four other invariants necessary for a symplectic classification, namely the total number of focus-focus points, the position of their images in the straightened image of the momentum map, a Birkhoff normal form type invariant (called 'Taylor series invariant') that describes the impact a focus-focus point has on the neighbourhood of its fibre, and, last but not least, the so-called twisting index invariant that describes the dynamical interaction between the different focus-focus fibres. Originally this classification applied only to semitoric systems with maximally one focus-focus point per fibre. Recently, Palmer & Pelayo & Tang [PPT] generalised the invariants to allow for more than one focus-focus point per fibre.
During the last decade, semitoric systems gained more and more attention and popularity. This is due to the fact that, although their classification is more involved than the one of toric systems, it is still 'doable and constructive': Pelayo & Vũ Ngo . c [PV2] showed, how to construct for given data that are admissible as invariants, a semitoric system having these data as invariants. They constructed the systems by gluing together coordinate patches containing the necessary singular points etc. But patching together plus smoothing along the seams of the patches is not very suitable for explicit observations how a semitoric system and its invariants change under variation of parameters.
For this, it is much more practical to have systems that are defined globally by one explicit formula on an explicitly given symplectic manifold. Recently, there has been progress in this direction: Alonso & Dullin & Hohloch [ADH1,ADH2] completed the computation of the classifying invariants for coupled spin oscillators with varying radius and coupled angular momenta with two varying radii and a coupling parameter. Hohloch & Palmer [HP] generalised the coupled angular momenta to an explicit semitoric family with two coupling parameters that has two distinct focus-focus points for certain intervals of the parameters. Le Floch & Palmer [LFP] pushed this further to explicit families of semitoric systems on Hirzebruch surfaces. By using invariant functions to perturb a given toric system, they came up with a method that seemed to work for more general situations than Hirzebruch surfaces.
This technique plus the question how difficult it is to construct explicit semitoric systems with more than two focus-focus points motivated the present paper: We construct a 1-parameter family of integrable systems F t that is at time t = 0 a toric system, then, as soon as t > 0, it becomes a system of toric type until some 0 < t − < 1 2 . At t = t − , four singular points undergo a Hamiltonian-Hopf bifurcation, changing from elliptic-elliptic to focus-focus. As soon as t > t − , the system is semitoric with four focus-focus points until some t + with 1 2 < t + < 1 where the focus-focus points undergo again a Hamiltonian-Hopf bifurcation that renders them elliptic-elliptic. Moreover, at time t = 1 2 , the focus-focus points team up in pairs of two that lie in the same focus-focus fibre, effectively creating thus two singular fibres that look like double pinched tori. For t + < t ≤ 1, the system is again of toric type.
The precise construction of F t goes as follows: First, starting with the octagon in Figure 1, we follow the steps of Delzant's [De] construction as described in detail in Cannas da Silva [CdS] to obtain a 4-dimensional, compact, connected, symplectic manifold (M, ω) := (M ∆ , ω ∆ ) by using symplectic reduction by a Hamiltonian T 6 -action of the 10dimensional preimage of a certain map from C 8 to R 6 (the details are given in Section 3). Points on (M, ω) are usually written as equivalence classes of the form [z] = [z 1 , . . . , z 8 ] with z k = x k + iy k ∈ C for 1 ≤ k ≤ 8. The momentum map of the toric system on (M, ω) is in fact surprisingly simple: This theorem is restated as Theorem 3.6 and proven throughout Section 3. In addition, in Proposition 3.7, we compute the precise coordinates of the eight elliptic-elliptic fixed points of F = (J, H). Now we look for a suitable function to perturb the integral H of F = (J, H) with. To this aim, consider the function Z : C 8 → C given by Z(z 1 , . . . , z 8 ) := z 2 z 3 z 4 z 6 z 7 z 8 . We will see in Lemma 4.3 that it descends to a function Z : (M, ω) → C and so do its real part R(Z) =: X and imaginary part I(Z) =: Y. Both functions are in addition invariant under J so that they also pass to the reduced spaces M red, j for j ∈ J(M). Now we vary H via linear combination (1 − 2t)H + tγX for γ > 0 sufficiently small and obtain Theorem 1.2. Let (M, ω, F = (J, H)) be the toric system from Theorem 1.1 and let 0 < γ < 1 48 and set F t := (J, H t ) := (J, (1 − 2t)H + tγX) : (M, ω) → R 2 . Then (M, ω, F t ) 0≤t≤1 is toric for t = 0, of toric type for 0 < t < t − , semitoric for t − < t < t + , and again of toric type for t + < t ≤ 1 where 0 < t − := 1 2(1 + 24γ) < 1 2 < t + := 1 2(1 − 24γ) < 1.
For all t ∈ [0, 1], the system F t has precisely eight fixed points of which four are always elliptic-elliptic. The other four pass at t = t − from elliptic-elliptic via a Hamiltonian-Hopf bifurcation to focus-focus. At t = t + , these four focus-focus points turn again back into elliptic-elliptic points via a Hamiltonian-Hopf bifurcation. For the special situation occuring at t = 1 2 , see Proposition 1.3. The momentum map image F t (M) is plotted for various values of t in Figure 2. Theorem 1.2 is restated as Theorem 4.7 in Section 4. Moreover, Proposition 4.8 computes the explicit coordinates of the eight fixed points of F t for time 0 ≤ t ≤ 1, thus extending Proposition 3.7 from the toric situation at t = 0 to t ∈ [0, 1].
Let us remark that, in the terminology of Kane & Palmer & Pelayo [KPP], the systems (F t ) t − <t<t + are minimal of type (6) with k = −2 and c = 4 and d = 4 (see [KPP,Theorem 2.4

and the table in Theorem 4.15]).
The proof of Theorem 1.2 resp. Theorem 4.7 is done in several steps spread over the following sections: • In Section 5, we show the existence and determine the positions of the fixed points.
• In Section 6, we show that the fixed points are nondegenerate and we determine their type. • In Section 7, we determine the rank one points and show that they are nondegenerate and of elliptic-regular type. • In Section 8, we summarise all steps and prove Theorem 4.7 and hereby also Theorem 1.2. At t = 1 2 , the fibres over (1, 0) and (2, 0) contain precisely two focus-focus points each. These fibres can be thought of as double pinched tori as sketched in Figure 3 and we exemplarily parametrise the one over (1, 0) explicitly. Proposition 1.3. At t = 1 2 , the system F 1 2 has precisely two focus-focus fibres, each of which contains precisely two focus-focus points so that each of these two fibres has the shape of a double pinched torus (see Figure 3). Exemplarily, F −1 This statement is reformulated as Proposition 4.9 and proven in Section 8.
The successful construction of the 1-parameter family F t suggests that Le Floch & Palmer's [LFP] method of interpolation with invariant functions may work in more generality. In particular, it should make the construction of semitoric systems possible with any wished number of focus-focus points and with a globally defined, explicitly given momentum map. In practice, the verification of such examples by hand may be lengthy and awkward since the number of equations defining the symplectic manifold grows.
Control of the Taylor series invariant or twisting index is unfortunately not (yet) possible with these techniques. Nevertheless, constructing focus-focus fibres containing two Figure 2. The image of the momentum map F t (M) of the family (M, ω, F t ) from Theorem 1.2 plotted with Mathematica for fifteen time steps between t = 0 and t = 1 with γ = 1 60 .
focus-focus points seemed easy enough. But since these focus-focus points originated from elliptic-elliptic points underlying vertices of the momentum polytope, one needs to employ additional methods if one wants to obtain fibres containing more than two focusfocus points. (1, 0) seen as double pinched torus. The parametrisation with r and ϑ in Proposition 1.3 attains A for r = 0 and B for r = √ 6. The sign in front of r tells the two 'bulges' of the double pinched torus apart. ϑ describes the rotational coordinate around each bulge. The plot is done with Mathematica.
. , X f n (p) are linearly independent for almost all p ∈ M. The function f is called the momentum map of the integrable system (M, ω, f ).
Let (M, ω, f ) be a 2n-dimensional completely integrable system. A point p ∈ M is called regular for f if X f 1 (p), . . . , X f n (p) are linearly independent. Otherwise p is said to be singular. The rank of the singular point p is the rank of (X f 1 (p), . . . , X f n (p)) or, equivalently, the rank of the Jacobian d f (p). A singular point of rank zero is usually called a fixed point. A value r ∈ R n is called regular if the whole fibre or level set f −1 (r) only contains regular points. Otherwise r ∈ R n is called singular. A fibre is said to be regular if it contains only regular points and otherwise singular.
2.3. Nondegeneracy and local normal form in 2n dimensions. A singular point p ∈ M of rank zero of a completely integrable system (M, ω, f = ( f 1 , . . . , f n )) is nondegenerate if the Hessians d 2 f 1 (p), . . . , d 2 f n (p) span a Cartan subalgebra of the Lie algebra of quadratic forms on T p M. We refer to Bolsinov & Fomenko [BF] for the precise definition of nondegenerate points of higher rank. For the case dim M = 4, we recall it below.
Williamson [Wi] classified Cartan subalgebras of sp(n, R). This in turn yields a classification of the possible subalgebras generated by the Hessians in T p M seen as sp(n, R). Vey [Ve], Eliasson [El1,El2], Miranda & Zung [MZ], Vũ Ngo . c & Wacheux [VuW], Chaperon [Ch], and others extended Williamson's pointwise classification to the following local classification and even more general versions.
Theorem 2.1 (Local normal form). Let p ∈ M be a nondegenerate singular point of a 2ndimensional completely integrable system (M, ω, f = ( f 1 , . . . , f n )). Then there exist local symplectic coordinates (x, y) := (x 1 , . . . , x n , y 1 , . . . , y n ) around p such that there exists q = (q 1 , . . . , q n ) : M → R n where each q j is given by one of (1) elliptic: q j (x, y) = 1 2 (x 2 j + y 2 j ), (2) hyperbolic: q j (x, y) = x j y j , (3) focus-focus: To detect the type of a nondegenerate singular point, it is sufficient to check the eigenvalues: Ngo . c[Vu2,Chapter 3]). Let C be a regular element in the Cartan subalgebra generated by the Hessians of the components of the momentum map (i.e., C has 2n distinct eigenvalues) at a fixed point. Then there appear three distinct types of groups of eigenvalues of C: (1) elliptic block: a pair of imaginary eigenvalues ±iβ, (2) hyperbolic block: a pair of real eigenvalues ±α, (3) focus-focus block: a quadruple of complex eigenvalues ±α ± iβ, where α, β ∈ R 0 .
2.4. Nondegeneracy and local normal form in 4 dimensions. In this paper, we mainly work on 4-dimensional symplectic manifolds, i.e., there are only singular points of rank zero or rank one possible. In this situation, nondegeneracy of rank zero points (= fixed points) can be verified as follows.
Nondegeneracy of rank one points can be characterised as follows on 4-dimensional symplectic manifolds (M, ω), see Bolsinov & Fomenko [BF,Section 1.8.2]. Let p be a singular point of rank one of a 4-dimensional completely integrable system M, ω, f = ( f 1 , f 2 ) . Then there are µ, λ ∈ R such that µ d f 1 (p) + λ d f 2 (p) = 0 and L p := Span{X f 1 (p), X f 2 (p)} ⊂ T p M is the tangent line in p of the orbit generated by the R 2action. Denote by L ⊥ p the symplectic orthogonal complement to L p in T p M. Note that L p ⊂ L ⊥ p . The Poisson commutativity { f 1 , f 2 } = 0 implies that they are invariant under the R 2 -action. Therefore µ d 2 f 1 (p) + λ d 2 f 2 (p) descends to the quotient L ⊥ p /L p . This allows to define Definition 2.4 (Bolsinov & Fomenko [BF]). A rank one critical point p of a 4dimensional completely integrable system M, The possible types of nondegenerate rank one points on 4-dimensional manifolds are • elliptic-regular if the eigenvalues of ω −1 p (µd 2 f 1 (p) + λd 2 f 2 (p)) on L ⊥ p /L p are of the form ±iα, α ∈ R 0 , • hyperbolic-regular if the eigenvalues of ω −1 p (µd 2 f 1 (p) + λd 2 f 2 (p)) on L ⊥ p /L p are of the form ±α, α ∈ R 0 . If one of the integrals has a periodic flow, the search for rank one points can be done by means of reduced spaces as described in Lemma 2.7 later on.
2.5. Symplectic reduction. Let us recall the following important notations and results: Theorem 2.5 (Marsden & Weinstein [MW]). Let G be a compact Lie group with Lie algebra g inducing a Hamiltonian action on a symplectic manifold (M, ω). Let f : M → g * be the momentum map of this action. Let r ∈ g * be a regular value of f that is fixed by the coadjoint action and denote by κ : f −1 (r) → M the inclusion. Assume that G acts freely and properly on f −1 (r). Then • Then there is a unique symplectic structure ω red,r on M red,r such that τ * ω red,r = κ * ω. The symplectic manifold (M red,r , ω red,r ) is called the symplectic reduction of M at level r by the action generated by G, also known as symplectic quotient or Marsden-Weinstein quotient.
We will use Theorem 2.5 (Marsden-Weinstein) in particular in the situation of a 4dimensional integrable systems where one of the integrals induces an S 1 -action: Corollary 2.6. Let (M, ω, F = (J, H)) be a 4-dimensional completely integrable system and let J have a periodic flow. Let j ∈ R be a regular value of J and assume that the S 1 -action induced by J is free and proper on J −1 ( j). Then the symplectic reduction of M at level j by the S 1 -action is given by the Marsden-Weinstein quotient M red, j := J −1 ( j)/S 1 and the Poisson commutativity of J and H assures that H descends to a smooth function H red, j : M red, j → R, called reduced Hamiltonian of H.  [HP,Corollary 2.5]. It means that the reduced Hamiltonian function is particularly useful to study singular points of rank one of the original system. Lemma 2.7. Let (M, ω, F = (J, H)) be a completely integrable system on a 4-dimensional symplectic manifold and let J have a periodic flow. Let j ∈ R be a regular value of J and assume that the S 1 -action induced by J is free and proper on J −1 ( j). Let p ∈ J −1 ( j) and set [p] := τ(p) ∈ M red, j . Then (1) p is a singular point of rank one of F = (J, H). ⇔ p is a singular point of H red, j , i.e., dH red, j ([p]) = 0.
(2) p is a nondegenerate singular point of rank one of F. ⇔ p is a nondegenerate singular point of H red, j . ⇔ dH red, j ([p]) = 0 and d 2 H red, j is invertible at p in M red, j . (3) p is an elliptic-regular (respectively hyperbolic-regular) point of F. ⇔ p is an elliptic (respectively hyperbolic) singular point of H red, j .
So far, we considered symplectic reduction of regular values. If the value is singular one encounters stratified spaces and needs to work with singular reduction, see Sjamaar & Lerman [SL] or, for an overview, Alonso [Al,, or do it by hand for simple examples as in Lemma 4.1.
2.6. Toric systems and Delzant's construction. Let (M, ω, f ) be a 2n-dimensional completely integrable system. If f is in fact the momentum map of a Hamiltonian n-torus action, we call (M, ω, f ) a toric manifold. According to the convexity theorem by Atiyah [At] and Guillemin & Sternberg [GuS], f (M) is a convex polytope spanned by the images of the fixed points. It is often referred to as momentum polytope. A convex polytope in R n is Delzant if (i) its edges have rational slope, (ii) at each vertex, precisely n edges meet, (iii) at each vertex, the n tangent directions considered as vectors in Z n span Z n .
Theorem 2.8 (Delzant [De]). A 2n-dimensional toric manifold having an effective Hamiltonian torus action is classified up to equivariant isomorphism by its momentum polytope which is in fact Delzant. Conversely, for each Delzant polytope, there exists, up to equivariant symplectomorphism, a toric manifold having the Delzant polytope as momentum polytope.
Crucial for the present paper is the fact that Delzant's proof is constructive, i.e., given a Delzant polytope ∆, there is an explicit way how to construct a toric manifold (M, ω, f ) with f (M) = ∆. We briefly sketch the construction as outlined in Cannas da Silva's book [CdS] since we will make ample use of it in Section 3: (1) Given a Delzant polytope ∆ ⊂ R n with k edges, write it as intersection of k halfspaces (whose boundaries lies on the edges of ∆) by means of the primitive, integral, outer normal vectors.
(2) Define a map ϑ : R k → R n by sending the standard basis of R n to the k outer normal vectors of the edges. This map is also welldefined as map ϑ : Z k → Z n and descends to ϑ : R k /Z k = T k → R n /Z n = T n . (3) The kernel N := ker(ϑ) of ϑ : T k → T n is isomorphic to an (k − n)-torus. Consider the inclusion N → T k and extend it to a map : R k−n → R n . Denote its dual map by * : R n → R k−n . (4) Let c ∈ Z k be the vector whose entries are given by the minimal distance to the origin of the boundaries of the halfspaces defining ∆. Consider the standard ktorus action on C k with momentum map L := L c : C k → R k , L c (z 1 , . . . , z k ) := − 1 2 |z 1 | 2 , . . . , |z k | 2 + c and define the mapL := * • L : C k → R k−n .
(5) Note that 0 ∈ R k−n is a regular value ofL and that N acts freely and properly oñ L −1 (0). According to Theorem 2.5 (Marsden-Weinstein), M red,0 :=L −1 (0)/N is a symplectic manifold with symplectic form ω red,0 which satisfies τ * ω red,0 = κ * ω st where τ :L −1 (0) →L −1 (0)/N is the quotient map and κ :L −1 (0) → (C k , ω st ) the inclusion and ω st the standard symplectic form on C k . (6) Find a right inverse σ : R n → R k of the map ϑ : R k → R n and consider the concatenationF SinceF is invariant under the action of N it descends to a map F : M red,0 → R n . Then F : (M red,0 , ω red,0 ) → R n is the momentum map of a Hamiltonian T n -action satisfying F(M red,0 ) = ∆.  [HSSS].
2.8. Semitoric systems and semitoric transition families. Toric systems only admit elliptic-elliptic or elliptic-regular points. In order to admit more types of singular points while keeping the class of systems as accessible as possible Vũ Ngo . c [Vu1,Vu3] began to focus on the following class of systems: 2) J is the momentum map of an effective Hamiltonian S 1 -action (i.e. the flow of X J is periodic, more precisely, 2π-periodic in our convention); 3) all singular points of F are nondegenerate and have no hyperbolic components.
A semitoric system is called simple if each fibre of J contains at most one focus-focus point. Simple semitoric systems where studied and classified by Pelayo and Vũ Ngo . c in [PV1,PV2] by means of five invariants.
The semitoric systems constructed later in Section 4 have always two focus-focus points in a fibre of J whenever the fibre contains focus-focus points, so they are not simple. However, had we started the construction with a Delzant octagon where the vertices of the horizontal edges do not lie on the same vertical line, then we would expect to get a simple semitoric system.
We will study the transition of elliptic-elliptic points to focus-focus points and back. Note that, at the moment of transition, the singular points have to be degenerate, i.e., the system is at this very moment not semitoric. The following definition is adapted from Le Floch & Palmer [LFP] who study transitions in parameter depending semitoric families over Hirzebruch surfaces.
Coupled angular momenta are an example for such a family. Hohloch & Palmer [HP] generalized them to a 2-parameter family admitting 2 focus-focus points while leaving the S 1 -action unchanged. Le Floch & Palmer [LFP] studied semitoric families with fixed S 1 -action on Hirzebruch surfaces.
Here we are in particular interested in fixed points changing from elliptic-elliptic to focus-focus and back (so-called Hamiltonian-Hopf bifurcations). The following definition is inspired by Le Floch & Palmer [LFP].
Definition 2.12. A semitoric family with transition points p 1 , . . . , p n ∈ M and transition times 0 < t − < t + < 0 is a semitoric family with fixed S 1 -action (M, ω, F t ) 0≤t≤1 and degenerate times t − and t + , such that 1) for 0 < t < t − and t + < t < 1, the points p 1 , . . . , p n are elliptic-elliptic; 2) for t − < t < t + , the points p 1 , . . . , p n are focus-focus; 3) for t = t − and t = t + , the points p 1 , . . . , p n are degenerate and there are no degenerate singular points in M \ {p 1 , . . . , p n }; 4) if p i is a maximum (resp. minimum) of H 0 | J −1 (J(p i )) then p i is a minimum (resp. maximum) of H 1 | J −1 (J(p i )) . Briefly, we just speak of a semitoric transition family in such a situation.
The following statements describe what happens at the transition times.
Proposition 2.13 (Hohloch & Palmer [HP]). Let (M, ω, F t ) 0≤t≤1 be a semitoric transition family with fixed S 1 -action. Let p ∈ M and t 0 ∈ ]0, 1[ be such that p is a fixed point for all t in a neighbourhood of t 0 where p is of focus-focus type for t > t 0 and of elliptic-elliptic type for t < t 0 . Then p is a degenerate fixed point for t = t 0 .
Note that singular points might change from rank zero to rank one points: Proposition 2.14 (Le Floch & Palmer [LFP]). Let (M, ω, (J, H t )) be a semitoric family with fixed S 1 -action and suppose that p ∈ M is a rank zero singular point for some t 0 ∈ [0, 1]. Then p is a singular point for all t ∈ [0, 1] but not necessarily of rank zero. If p does not belong to a fixed surface of J, then p is a fixed point for all t ∈ [0, 1].
3. The toric system constructed from the octagon in Figure 1 Using Delzant's construction sketched in Section 2.6, we will construct in this section the toric system associated to the polytope ∆ in Figure 1.
Note that, since is injective, * is surjective. The goal is to consider N as an action on C 8 and then to pass to the quotient. As a first step, we note This action is Hamiltonian and admits the family of momentum maps Now choose c = (0, −1, 0, 2, 3, 5, 3, 2) which are the distances of the eight half planes in (1) to the origin and set L := L (0,−1,0,2,3,5,3,2) : C 8 → R 8 .
Recall T 6 N ⊂ T 8 and use the inclusion N → T 8 as motivation to define the action The corresponding momentum map is given by which is in coordinates given by Note that, for the action of N on (C 8 , ω 0 ) with momentum map L : C 8 → R 6 , the value 0 ∈ R 6 is regular since * is surjective. The regular level set L −1 (0) is described by six equations: (4) are referred to as the manifold equations of M. Consider the inclusion κ : L −1 (0) → C 8 and note that N acts freely and properly on L −1 (0) (for a proof, see Cannas da Silva [CdS]). Therefore we may apply Theorem 2.5 (Marsden-Weinstein) to this situation and conclude that is a 4-dimensional symplectic manifold, denoting the quotient map by τ : L −1 (0) → M, On the quotient, we have the symplectic form ω := ω red,0 which satisfies Explicit charts of the manifold M. For the constructions later on, we need explicit charts for M and some additional notation.
Notation 3.2. The notation mod 8 means that integers are counted modulo 8 and that they are considered as element of the set {1, 2, 3, 4, 5, 6, 7, 8}. Note that this set starts with 1 instead of 0.
To define suitable charts for the manifold M, we will make use of the following technical observation.
Then z j 0 for j {k − 1, k + 1} mod 8. This means that maximally two coordinates of z may vanish and, if so, these must be consecutive ones.
Proof. We show the assertion for k = 1. The other cases follow similarly. Let z 1 = 0 and consider the manifold equations (A) -(F) from (4). Equation (A) leads to |z 5 | 2 = 6 0 implying z 5 0. That z 3 , z 4 , z 6 , z 7 must be nonzero is proven by contradiction: Thus, if z 1 = 0, only z 2 or z 8 may vanish, too. Moreover, since the action of N on L −1 (0) preserves the norm |z j | for each entry 1 ≤ j ≤ 8, the action does not affect the property of being zero.
This motivates us to define, for 1 ≤ ν ≤ 8, the open sets In other words, U ν is the only subset of M where z ν and z ν+1 may vanish. This is independent of the chosen representative for the point [z 1 , . . . , z 8 ] ∈ M. In particular, we have an open covering M = 8 ν=1 U ν . Now we will construct explicit charts ψ ν : U ν → R 4 for all 1 ≤ ν ≤ 8. Given [z] ∈ U ν , there are at least six variables nonzero among z 1 , . . . , z 8 . Now use the N T 6 -action to pick strictly positive real numbers as representatives for them. Writing z k = x k + iy k for 1 ≤ k ≤ 8, this means y k = 0 and x k > 0 for these six representatives. Therefore, for ν = 1, we represent U 1 by points of the form Using the manifold equations from (4), the variables x 3 , . . . , x 8 can be written in U 1 as functions of x 1 , y 1 , x 2 , y 2 as follows: abbreviating |z 1 | 2 = x 2 1 + y 2 1 and |z 2 | 2 = x 2 2 + y 2 2 , we obtain Therefore is a chart for U 1 with inverse where x 3 , . . . , x 8 in ψ −1 1 are determined by (6). Applying analogous reasoning to the remaining indices, we obtain, for all 1 ≤ ν ≤ 8, charts of the form 3.3. The symplectic form ω on M in local coordinates. In this subsection, we will compute the symplectic form ω on M constructed in Section 3.1 in the local charts (U ν , ψ ν ) defined in Section 3.2. Let defined analogously to the chart ψ ν , ψ −1 ν in Section 3.2, i.e., we have τ • τ −1 ν = Id U ν and thus in turn Consider the symplectic form ω st on C 8 R 16 as represented by the (16 × 16)-matrix with submatrices 0 −1 1 0 on the diagonal and everywhere else zero. For ν = 1, we now compute the pullback ω 1 of the matrix of ω st explicitly. The cases ν = 2, . . . , 8 are similar. We find and calculate the Jacobian which, considering ω st and ω 1 as matrices, yields 3.4. Momentum map of the toric system on (M, ω). Now let us resume Delzant's construction (see Section 2.6 for an overview of the steps) to obtain a momentum map F on M satisfying F(M) = ∆. First, we have to find a map σ : R 2 → R 8 that is a right inverse of the map ϑ : Lemma 3.4. The map σ := 1 6 ϑ * : R 2 → R 8 is a right inverse of ϑ. Proof. Written in matrix form, ϑ : R 8 → R 2 is given by Therefore the dual map is given by ϑ * : R 2 → R 8 , ϑ * (y) = C T y. Since CC T = 6 Id R 2 , we find Id R 2 = 1 6 ϑ • ϑ * = ϑ • 1 6 ϑ * such that σ := 1 6 ϑ * is a right inverse of ϑ. The next step in Delzant's construction is to compute the concatenation where we remark that σ * = 1 6 ϑ.
is a momentum map of an effective Hamiltonian 2-torus action and satisfies F(M) = ∆. The S 1 -action of J is given by and the S 1 -action of H by Proof. Constructed via Delzant's construction, F is the momentum map of an effective Hamiltonian 2-torus action. Note that {J, H} = 0 is actually additionally proven, explicitly in local coordinates, in the proof of Proposition 4.6 when showing Poisson commutativity of the integrals of the (future) semitoric system.
Let us now verify F(M) = ∆. Recall the manifold equations (A) -(F) from (4) and note that equations (A) and (C) imply 0 ≤ J ≤ 3 and 0 ≤ H ≤ 3 and |z 5 | 2 = 6 − 2J which we will use in the following. We conclude   Proposition 3.7. F = (J, H) has precisely the following eight fixed points: is the fixed point with z 1 = z 2 = 0, is the fixed point with z 2 = z 3 = 0, is the fixed point with z 3 = z 4 = 0, is the fixed point with z 4 = z 5 = 0, is the fixed point with z 5 = z 6 = 0, is the fixed point with z 6 = z 7 = 0, is the fixed point with z 7 = z 8 = 0, is the fixed point with z 8 = z 1 = 0.
They are mapped under F to the vertices of the octagon as displayed in Figure 4.
Proof. We first look for the fixed points of J and H separately since fixed points of F = (J, H) are precisely the points that are fixed points for both J and H. Fixed points of J: The point [z] = [z 1 , . . . , z 8 ] ∈ M will be fixed by the action of J if the following holds true: Case 1: z 1 = 0 in the first coordinate yields a fixed point of the J-action. Case 2: z 5 = 0 also gives rise to a fixed point, since we can take (t 1 , . . . , t 6 ) = (−t, 0, . . . , 0) in the N-action, which only gives another representation for the same point in M. This means we can actually see the J-action as a special case of the N-action.
Case 3: Seeing the J-action as an N-action with (t 1 , . . . , t 6 ) = (−t, 0, . . . , 0) means that t rotates both z 1 and z 5 . There are several possibilities to compensate the rotation of z 5 . Let us for example set the parameter t 2 = t (so that z 5 is kept invariant). Then t 2 also affects z 2 and z 7 , so a fixed point should satisfy z 2 = 0 and either z 7 = 0 or one of the z 3 , z 4 , z 6 , z 8 is zero, where we choose a third parameter (for example t 3 = −t) in the N-action. We know from Lemma 3.3 that only two subsequent entries can be zero, so the only remaining possibility is z 2 = 0 and z 3 = 0.
Case 4: The same idea works when we compensate the rotation of z 5 by choosing another parameter in the N-action: picking t 4 = −t leads to z 4 = 0 and z 3 = 0. Moreover, setting t 5 = −t gives rise to z 6 = 0 and z 7 = 0. Last, considering t 6 = t implies z 8 = 0 and z 7 = 0.
Fixed points of H: Similar to the previous case, a point [z] = [z 1 , . . . , z 8 ] will be fixed by the H-action in the following cases: Case 1: z 3 = 0. Case 2: z 7 = 0. Case 3: z 2 = 0 and z 1 = 0; z 4 = 0 and z 5 = 0; z 6 = 0 and z 5 = 0; z 8 = 0 and z 1 = 0. Conclusion: When considering these conditions together, we see that the fixed points are precisely those points with two subsequent entries equal to zero. If we consider such a fixed point with z ν = 0 and z ν+1 = 0, then this point lies in U ν ⊆ M and can be parametrised by (x ν , y ν , x ν+1 , y ν+1 ) = (0, 0, 0, 0). All other entries can be written as positive real numbers and are found by means of the manifold equations.

A semitoric family with four focus-focus points and two double pinched tori
Within this section, let (M, ω, F = (J, H)) be the toric manifold constructed in Section 3 that satisfies F(M) = ∆ where ∆ is the octagon from Figure 1.
The aim of this section is to replace the second integral H by a parameter depending family H t such that (M, ω, F t = (J, H t )) is a semitoric family with fixed S 1 -action having four transition points A, B, C, D ∈ M which map under F 0 = (J, H 0 ) to the points (1, 3), (1, 0), (2, 3), (2, 0) as sketched in Figure 4. 4.1. Geometric interpretation of M red, j . Following Karshon [Ka], we call a value j of J extremal if j is a global minimum or maximum of J, and interior otherwise. A fixed point of J in the fibre J −1 ( j) is extremal if j is extremal. Analogously we define interior fixed points. The following statement is a compilation of Lemma 2.12 in Le Floch & Palmer [LFP] and Proposition 3.4 in Hohloch & Sabatini & Sepe [HSS].
Lemma 4.1. Let (M, ω, F = (J, H)) be a semitoric system. 1) Let j ∈ J(M) be an interior value. If j is regular for J, then M red, j is diffeomorphic to a 2-sphere. If j is singular for J then M red, j is homeomorphic to a 2-sphere (but not diffeomorphic). 2) Let j ∈ J(M) be an extremal value of J. If j corresponds to a vertical edge of the momentum polytope of F = (J, H) then M red, j is diffeomorphic to a 2-sphere. Otherwise M red, j is a point.
A look at Figure 1 shows that j ∈ {0, 3} are the extremal values of J and j ∈ {1, 2} singular interior values. All j ∈ [0, 3] \ {0, 1, 2, 3} are regular interior values. We conclude The situation is displayed in Figure 5. Let us explain the geometric intuition of the shape of the reduced spaces and positions of the singular points of H red, j . Later in Section 4.2, we will give a parametrisation of M red, j .
Recall that, according to Lemma 2.7, nondegenerate elliptic-regular points of F = (J, H) on the 4-dimensional manifold (M, ω) that are regular for J correspond to nondegenerate elliptic points of H red, j on the 2-dimensional space M red, j . Hence there are, for j ∈ ]0, 3[ \{1, 2}, precisely two elliptic points of H red, j on M red, j . They are located at the 'north and south pole' of M red, j di f f eo S 2 since, as a maximum and minimum of H red, j , the north and south pole are elliptic points of H red, j . But there are not more than two elliptic points possible for H red, j since there are only two elliptic-regular points in F −1 ( j). Now consider the interior singular values j ∈ {1, 2}. The proof of Lemma 2.12 in Le Floch & Palmer [LFP] (see also the overview in Alonso [Al,] on singular reduction) shows that an elliptic-elliptic or focus-focus point on M causes M red, j to have a singularity (a 'non-smooth peak') such that M red, j is only homeomorphic to a 2-sphere but not diffeomorphic. Since there are two elliptic-elliptic points in the level set of j ∈ {1, 2} of J, the reduced space M red, j has two 'non-smooth peaks' as displayed in Figure 5. Now consider the extremal values j ∈ {0, 3} where, according to Lemma 4.1, the reduced space is diffeomorphic to a 2-sphere. Thus there are two (elliptic) points Figure 5. M red, j plotted for j ∈ {0, 0.5, 1, 1.5, 2, 2.5, 3} with Mathematica. For j = 1 and j = 2, the reduced space M red, j contains two singular points, i.e., it is homeomorphic to a 2-sphere with two 'peaks', otherwise M red, j is diffeomorphic to a 2-sphere.
[p max ], [p min ] ∈ M red, j where H red, j attains its maximum and minimum, i.e., dH red, j vanishes in [p max ] and [p min ]. Since j is extremal, we have dJ(p) = 0 for all p ∈ J −1 ( j). Altogether we conclude that [p max ] and [p min ] are mapped to elliptic-elliptic points of F = (J, H) which are precisely the upper and lower vertex of the vertical edges over j = 0 and j = 3 in the octagon ∆. The other points in M red, j correspond to elliptic-regular rank one points of M and are mapped to the vertical segments of the momentum polytope between the vertices. 4.2. Parametrisation of M red, j . After these geometric considerations, we explain now how to obtain the coordinates used for displaying the reduced spaces M red, j in Figure 5. The idea is to 'foliate' the 2-dimensional space M red, j by the level sets of H red, j and consider H red, j as a height function of M red, j . Then the possible values h j ∈ H red, j (M red, j ) lead to one coordinate. The other coordinate comes from parametrising the (usually 1dimensional) level sets of H red, j . Herefore we need a map that is welldefined on the reduced spaces M red, j and whose level sets are transverse to the ones of H red, j . are N-invariant and thus descend as welldefined functions to M. Moreover, X and Y are also J-invariant and therefore descend to the reduced spaces M red, j for j ∈ J(M).
By means of the implicit function theorem, we would like to use Z = (X, Y) to obtain coordinates on M red, j of the form u, H red, j (u) or u −1 (H red, j ), H red, j . Thus we need to relate Z = (X, Y) to H and J. Proof. Use the definition of J and H to find |z 1 | 2 = 2J and |z 3 | 2 = 2H and use the manifold equations to get furthermore |z 2 | 2 = −2 + 2H + 2J, |z 5 | 2 = 6 − 2J, |z 7 | 2 = 6 − 2H, We now calculate Thus Z = (X, Y) is related to J and H via a rotation invariant formula. Therefore we may set Y = 0, solve for X and obtain M red, j parametrised as a surface of revolution in R 3 with coordinates (X, Y, H red, j ). Then, for fixed j ∈ J(M), the reduced space M red, j can be parametrised by The function h j → ±X( j, h j ) is plotted for various values of j in Figure 6. Rotating the image of h j → X( j, h j ) around the horizontal axis gives the space M red, j that is plotted in Figure 5 in the (rotated) coordinate system (X, Y, H red, j ).  4.3. The family F t = (J, H t ) of integrable systems. We know from Theorem 3.6 that (M, ω, F = (J, H)) is toric. The idea is now to obtain a family H t by interpolation between H and X, i.e., we consider a new 'height function' that changes from H to X when varying the parameter. Later on in this paper, we will need (parts of) the explicit calculations in local coordinates of {J, H t } = 0, so we add here here the proof in local coordinates. We only consider the chart (U 1 , ψ 1 ) since the cases 2 ≤ ν ≤ 8 go analogously. We find and, using x 3 = 2 − |z 1 | 2 + |z 2 | 2 from (6) and y 3 = 0, we get . Recall that the symplectic form ω 1 on ψ 1 (U 1 ) = V 1 is the standard symplectic form ω st on R 4 . We calculate X J•ψ −1 1 (x 1 , y 1 , x 2 , y 2 ) = (y 1 , −x 1 , 0, 0) T , Thus J and H Poisson commute. Moreover, using the definition of ψ −1 1 , we find and by using the relations defining x 3 , . . . , x 8 we get (X • ψ −1 1 )(x 1 , y 1 , x 2 , y 2 ) = x 2 x 3 x 4 x 6 x 7 x 8 (9) = x 2 2 − (x 2 1 + y 2 1 ) + (x 2 2 + y 2 2 ) 6 − 2(x 2 1 + y 2 1 ) + (x 2 2 + y 2 2 ) 8 − (x 2 2 + y 2 2 ) 4 + (x 2 1 + y 2 1 ) − (x 2 2 + y 2 2 ) 2 + 2(x 2 1 + y 2 1 ) − (x 2 2 + y 2 2 ) which can be seen, in the first two variables, as a function depending on ξ 1 = x 2 1 and η 1 = y 2 1 and that is symmetric in ξ 1 and η 1 . Thus the partial derivatives of X • ψ −1 1 in (x 1 , y 1 , x 2 , y 2 ) w.r.t. x 1 and y 1 coincide except for the factor x 1 resp. y 1 , i.e., we obtain d(X • ψ −1 1 ) = x 1 f, y 1 f, g, h for suitable functions f , g, h : V 1 → R with coordinates (x 1 , y 1 , x 2 , y 2 ). This yields at the point (x 1 , y 1 , x 2 , y 2 ) Since H t = (1 − 2t)H + tγX, linearity of the Poisson bracket yields {J • ψ −1 1 , H t • ψ −1 1 } = 0 for all t and all γ. Now we show that X J and X H t are almost everywhere linearly independent. For t = 0, we have H 0 = H and thus the claim follows from F = (J, H) being an integrable system.
. This is only possible if y 2 = 0 and x 2 satisfies an equation, so these (x 1 , y 1 , x 2 , y 2 ) live in a two-dimensional subset and hence, the linear independence holds almost everywhere.

4.4.
Intuition on focus-focus points of the family F t = (J, H t ). On the reduced space, the family H t has the following geometric meaning: For t = 0, we have H red, j 0 = H red, j which is our 'vertical' height coordinate. For t ∈ ]0, 1 2 [, the 'height' H red, j t is measured 'tilted more and more towards the horizontal' X-coordinate which is reached (up to scaling by γ 2 ) at t = 1 2 with H red, j 1 2 = ( γ 2 X) red, j . Then, the singular points of the manifold are interior points of this function and will be singular points of focus-focus type. This idea is visualised in Figure 7 for the level J = 1 = j. = ( γ 2 X) red,1 and the singular level is the blue curve connecting the two singular focus-focus points. This plot is done with Mathematica.

4.5.
Intuition on the rank 1 points of the family F t = (J, H t ). At the interior levels j ∈ ]0, 3[ \{1, 2}, the reduced space M red, j is diffeomorphic to a 2-sphere. Thus, no matter how far H j,red t is 'tilted' from H towards X, the maximum and minimum of H red, j t are each reached at a unique point on M red, j .
By Lemma 2.7, these points correspond to the rank one points of (M, ω, F t ) which will turn out to be nondegenerate of elliptic-regular type and map to the top and bottom edge points in the momentum polytope F t (M). As H red, j t 'tilts' from H red, j = H to X as t increases, also the rank one points on M may change.
For the levels j ∈ {0, 3}, the minimum resp. maximum of H red, j t is also attained at a unique point on M red, j . Both correspond to rank zero points of elliptic-elliptic type on M.
When t changes from 0 to 1, other rank one points will become the rank zero points, but, according to Proposition 2.14, none of these points ever becomes regular.
For the levels j ∈ {1, 2}, the reduced manifold has two singular points, which are the maximum and minimum of H red, j t as long as t < t − and t + < t for certain 'degenerate times' 0 < t − < 1 2 < t + < 1. When t is in between the degenerate times, two other points on the reduced space will become the extremal values of H red, j t . These points correspond to nondegenerate elliptic-regular rank one points on M which are mapped to the upper and lower boundary of the octagon. The fixed points are then focus-focus points and mapped to the interior. 4.6. Main results. We showed in Theorem 3.6 that (M, ω, F = (J, H)) is a toric system with F(M) = ∆. The family F t = (J, H t ) was shown to be an integrable system in Proposition 4.6. But (M, ω, F t ) has even nicer properties: Theorem 4.7. Let (M, ω, F = (J, H)) be the toric system from Theorem 3.6 and let M, ω, F t = (J, (1 − 2t)H + tγX) be the family of integrable systems from Proposition 4.6 and let 0 < γ < 1 48 . Then (M, ω, F t ) 0≤t≤1 is a semitoric transition family with fixed S 1 -action having four transition points and two transition times The proof is spread over the following sections and summarised in Section 8. The image F t (M) of the momentum map is plotted for γ = 1 60 in Figure 2: The images of four elliptic-elliptic fixed points 'pass into the interior' of the momentum polytope at t = t − , becoming focus-focus points. At t = 1 2 , the focus-focus points form two pairs where each pair is mapped to the same value in the momentum polytope, i.e., the points of a pair lie in the same fibre. At t = t + , the images of the focus-focus points become again boundary points of the momentum polytope, i.e., they switch back to being elliptic-elliptic. We compute now the coordinates in M of the eight fixed points of F t = (J, H t ).
Proposition 4.8. For all t ∈ [0, 1], the system F t = (J, H t ) has precisely eight fixed points. They are given by four points not depending on t, namely as in Proposition 3.7, and four points that change with t as follows: √ 2] are the unique smooth solutions of the equation This will be proven in Section 5. Let us now shed some light on the situation at t = 1 2 . Proposition 4.9. At t = 1 2 , the focus-focus points A and B lie both in F −1 1 2 (1, 0) and the focus-focus points C and D lie both in F −1 1 2 (2, 0) and both fibres have the form of a double pinched torus as displayed in Figure 3. Exemplarily, we compute the fibre F −1 1 2 (1, 0) as Using polar coordinates, this yields for F −1 This statement will be proven in Section 8.

The whereabouts of the fixed points of F t = (J, H t )
In this section, we will determine the explicit coordinates of the eight fixed points of F t = (J, H t ) and hereby prove Proposition 4.8.
We only have to consider the case t > 0 since t = 0 is already treated in Proposition 3.7. For t > 0, the family F t = (J, H t ) will turn out to be of toric type or semitoric -apart from the two transition times where the four transition points pass through a degeneracy.
Proof of Proposition 4.8. p = [p 1 , . . . , p 8 ] ∈ M can only be a fixed point of F t = (J, H t ), if it is a fixed point of J. Thus, according to Proposition 3.7, p lies in {A, B, C, D} or it satisfies p 1 = 0 or p 5 = 0.
Using the definition of ψ −1 1 , i.e., the formulas in (6), we obtain the coordinates of P min t and P max t as given in Proposition 4.8. Letting t → 0, we recover P max = P max 0 and P min = P min 0 of the toric system, as computed in Proposition 3.7.
Then the rank zero points A, B, C, D are of elliptic-elliptic type for 0 ≤ t < t − as well as for t + < t ≤ 1 and of focus-focus type for t − < t < t + . Moreover, for t = t ± , the points A, B, C, D are degenerate (see Proposition 2.13). Proof.
Consider the case 0 < t < t − or t + < t ≤ 1: Here we have δ > 0 so that χ has two real roots We haveb and t < t − or t + < t. Hence we find λ − < 0. Moreover,b > √ δ since b 2 − ∆ = 4c 4 t 4 > 0. This implies λ + < 0 as well. Therefore, the zeros of χ are of the form ±iα, ±iβ and A is nondegenerate of elliptic-elliptic type according to the list after Lemma 2.3.
Proof for the points C and D: Analogous to the ones for A and B.
We now study the type of the remaining four fixed points. Proof. We showed in Proposition 4.8 that we recover via P min 0 = P min , P max 0 = P max , Q min 0 = Q min , Q max 0 = Q max the fixed points of the toric system (M, ω, F = (J, H)) which are all elliptic-elliptic. Thus it is sufficient to prove Proposition 6.2 for t > 0.

M(u)
, we obtain We calculate the type of the singular points by finding the eigenvalues of the matrix The characteristic polynomial is given by Hence the eigenvalues are of the form λ ± = ±ia(u) and µ ± = ±i b(u)c(u).
Now we show that λ ± and µ ± are purely imaginary. Let us first investigate λ ± . Since u ∈ ] − √ 2, √ 2 [ for t > 0, we have M(u) > 0 so that M(u) and a(u) are real numbers. Since u = u ± (t) is also real valued, λ ± = λ ± t is purely imaginary for all t ∈ ]0, 1]. Now we investigate µ ± . First, recall the function f from (10) and note that f(u 2 ) = M(u) and 2uf (u 2 ) = M (u). Thus we can rewrite The nominator γt M(u) is a strict positive number for t > 0 and hence, we conclude that c(u) and u have opposite signs. Moreover, .
Hence b(u) and u have opposite signs. Altogether we conclude b(u)c(u) > 0 and thus µ ± = µ ± t is purely imaginary for all t ∈ [0, 1]. Finally, a look at the list after Proposition 4.8 shows that P min t and P max t are nondegenerate points of elliptic-elliptic type for all t ∈ [0, 1].
Note thatã(u) coincides with a(u) except for the minus sign at the term (1 − 2t) right at the beginning. The same holds true forb(u) and b(u) andc(u) and c(u). Thus we obtain almost the same matrix up to these terms. Now use (11) instead of (10), where this opposite sign also appears. Similar calculations as above show that Q min t and Q max t are also nondegenerate singular points of elliptic-elliptic type.

The singular points of rank one and their types
For t = 0, the system F 0 = (J, H 0 ) = (J, H) is toric (see Theorem 3.6) so that its rank one points are elliptic-regular and mapped to the edges of the momentum polytope F 0 (M) = ∆. In what follows, we will study the case t > 0: we will first investigate the case dJ(p) 0 and then the case dJ(p) = 0. 7.1. Case dJ(p) 0. Geometrically this means that, at such a rank one point p, either dH t (p) = 0 or dH t (p) and dJ(p) are linearly dependent.
Suppose now that p 2 = 0. Since t 0 and γ > 0, it follows from the second equation that at least one of p 3 , p 4 , p 6 , p 7 , p 8 must be zero. A similar conclusion is true if we assume one of p 3 , p 4 , p 6 , p 7 , p 8 to vanish. Lemma 3.3 implies that never more than two subsequent coordinates p k for k ∈ {2, 3, 4, 6, 7, 8} are zero. But in this case, we get one of the fixed points A, B, C, D which are rank zero points instead of rank one points with dJ(p) = 0. Thus a singular point p of rank one with dJ(p) 0 has entries p 2 , p 3 , p 4 , p 6 , p 7 , p 8 0.
Recall from the proof of Proposition 3.7 that the set of points with dJ = 0 consists of the fixed points A, B, C, D together with all points z satisfying z 1 = 0 or z 5 = 0. Thus, if dJ(z) 0 for a point z ∈ M, we have z 1 , z 5 0. Together with Lemma 7.1, this means that the coordinate entries of a singular point of rank one never vanish. W.l.o.g. let us work in the chart (U 1 , ψ 1 ). Given z ∈ U 1 with dJ(z) 0 (and thus no entries equal to zero), write it as z = ψ −1 1 (x 1 , y 1 , x 2 , y 2 ) = [x 1 , y 1 , x 2 , y 2 , x 3 , 0, . . . , x 8 , 0] and rotate z 1 = x 1 + iy 1 with the S 1 -action of J until y 1 = 0. Abbreviate J(z) =: j and note that we have in this situation x 1 = 2 j. Now consider the image of z under the quotient map to the reduced space M red, j . Note that the possible choices of z 2 = x 2 + iy 2 in the chart z = ψ −1 1 (x 1 , y 1 , x 2 , y 2 ) now describe M red, j completely. Since z 2 0, we may write z 2 = ρe iϑ in polar coordinates. We have ϑ ∈ [0, 2π[, but the actual values of ρ > 0 are implicitly determined by the manifold equations (4) which read in the new coordinates x 4 = 6 − 4 j + ρ 2 , y 1 = y 3 = y 4 = y 5 = y 6 = y 7 = y 8 = 0.
This change of coordinates is of use for  The 'orange peaks' rise at the boundary of the admissible region. On the boundary itself, the function may vanish (e.g. f (0, 1) = 0). All three plots are realised with Mathematica.
7.2. Case dJ(p) = 0. Geometrically this means that, at such a rank one point p, we have dH t (p) 0, i.e., we are dealing with points that are fixed under the flow of J, but are non-fixed by the flow of H t . Recall from the proof of Proposition 3.7 that the set of points with dJ = 0 consists of the fixed points A, B, C, D together with all points z satisfying z 1 = 0 or z 5 = 0. Proposition 7.3. For t ∈ ]0, 1], the rank one points of F t = (J, H t ) in J −1 (0) ∪ J −1 (3) are nondegenerate of elliptic-regular type .
Therefore, we get Therefore ω −1 1 d 2 (J • ψ −1 1 ) descends to L ⊥ p /L p as 0 1 −1 0 which has purely imaginary eigenvalues ±i. Thus, by Definition 2.4 and the discussion afterward, the rank one points are nondegenerate of elliptic-regular type.
The chart (U 5 , ψ 5 ) contains all possible rank one points in J −1 (3) except those with 0 = p 4 for which we have to use the chart (U 4 , ψ 4 ). Analogous calculations as above show that there is only one such point and that it is of elliptic-regular type.
8. The proofs of Theorem 4.7 and Proposition 4.9 Proof of Theorem 4.7. In Theorem 3.6, we showed that (M, ω) is a 4-dimensional compact symplectic manifold and that F = (J, H) is, up to equivariant symplectomorphism, the toric system satisfying F(M) = ∆ where ∆ is the octagon from Figure 1. In Proposition 4.6, we extended (M, ω, F = (J, H)) to a family (M, ω, F t = (J, H t )) of integrable systems with F 0 = F. Since J does not depend on t, its induced S 1 -action is unchanged under variation of t. Moreover, since M is compact, J : M → R is proper.
It remains to show that, apart from a finite number of transition times, the system F t = (J, H t ) is semitoric, i.e., singularities are nondegenerate and have no hyperbolic components and that, at the transition times, certain singular points change from ellipticelliptic to focus-focus or vice versa.
In Section 5, we deduced the eight fixed points of F t = (J, H t ) for t ∈ ]0, 1] denoted by A, B, C, D, P min t , P max t , Q min t , Q max t and their coordinates, hereby proving Proposition 4.8. In Proposition 6.1, we showed that there are two transition times 0 < t − < 1 2 < t + < 1 where A, B, C, D switch from being nondegenerate and elliptic-elliptic via a degeneracy at t − to being nondegenerate and focus-focus and then via a degeneracy at t + to being nondegenerate and elliptic-elliptic. In Proposition 6.2, we showed that P min t , P max t , Q min t , Q max t are nondegenerate and elliptic-elliptic for all t ∈ ]0, 1]. Proposition 7.2 proves that the rank one points of F t = (J, H t ) on M \ (J −1 (0) ∪ J −1 (3)) are nondegenerate and of elliptic-regular type for all t ∈ ]0, 1]. Proposition 7.3 shows that the rank one points in J −1 (0) and J −1 (3) are nondegenerate and of elliptic-regular type for all t ∈ ]0, 1]. This finishes the proof of Theorem 4.7.
It remains to prove Proposition 4.9, i.e., that, at t = 1 2 , the focus-focus points A and B lie both in F −1 1 2 (1, 0) and that the focus-focus points C and D lie both in F −1 1 2 (2, 0) and that this gives the fibres the form of a double pinched torus.
Proof of Proposition 4.9. Zung [Zu1,Zu] showed that a focus-focus fibre containing exactly n focus-focus points consists of a chain of n spheres where each of the spheres intersects transversally two others and that the intersection points are given by the n focusfocus points. Such a fibre is said to have the form of an n-pinched torus. In the special case n = 1, the fibre is a sphere with one point of self-intersection and is said to be a single pinched torus.
We now parametrise the fibre F −1 1 2 (1, 0) as follows. Given any z = [z 1 , . . . , z 8 ] ∈ F −1 1 2 (1, 0) ⊂ M, we have J(z) = 1 and thus |z 1 | 2 = 2. Moreover, we find 0 = X(z) = Concerning the reformulation in polar coordinates, we remark that r = 0 recovers the point A and r = √ 6 gives the point B. For all other values of r, the fibre splits up in two parts, one for the plus and one for the minus sign in the seventh coordinate. The rotation related to the S 1 -action induced by J is described by the parameter ϑ, so indeed, we get a 2-torus pinched at the points A and B, as visualised in Figure 3.