Classification of Hamiltonian non-abelian Painlev\'e type systems

All Hamiltonian non-abelian Painlev\'e systems of ${\rm{P}}_{1}-{\rm{P}}_{6}$ type with constant coefficients are found. For ${\rm{P}}_{1}-{\rm{P}}_{5}$ systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new ${\rm{P}}_{3}^{\prime}$ and ${\rm{P}}_{5}$ systems thus obtained, we find isomonodromic Lax pairs for them.


Introduction
In the paper [7] by K.Okamoto all Painlevé equations P 1 − P 6 were written as polynomial Hamiltonian systems of the form The Okamoto Hamiltonian for the i-th Painlevé system has the form 1 fi(z) h i , where Here κ 1 , κ 2 , κ 3 , κ 4 are arbitrary constants.Note that since the systems are non-autonomous, the Hamiltonians are not integrals of motion.
In [4] H. Kawakami constructed Hamiltonian matrix generalizations of these Painlevé P 1 − P 6 systems.The corresponding Hamiltonian functions have the form 1 fi(z) trace (H i ), where H i are (non-commutative) polynomials with constant coefficients in two matrices u and v, linear in z.If the size of matrices is equal to one, these Hamiltonians coincide with the Okamoto's ones.
In this paper we never use matrix entries, but operate only with polynomials in noncommutative variables u, v.More rigorously, we are dealing with ODEs in free associative algebra A = C[u, v] with the unity 1.The independent variable z plays here the role of a parameter.The corresponding definitions of trace functional and non-abelian partial derivatives used in formula (1) are given in Section 2 (see also [5]).
In Section 3 we find all Hamiltonian non-abelian systems of Painlevé type.Our classification is based on the following assumption: Assumption 1.For each non-abelian Painlevé system (1) of type k, k = 1, ..., 6 there exist polynomials is the Hamiltonian of the system;

the scalar reductions of polynomials S (i)
k coincide with the powers h i k ;

trace S (i)
k and trace S (j) k commute with each other with respect to the symplectic nonabelian Poisson bracket (see Section 2) for any i, j.
In fact, the result of our classification coincides with the collection of the Kawakami systems.A small generalization is the presence of the additional parameters β, γ in the systems P ′ 3 , P 5 , P 6 .Note that the Kawakami systems are not invariant under the simplest Bäcklund transformations (see Appendix A.1), since these transformations change not only the parameters κ i , but also β, γ.
The parameters β, γ are not essential in the following sense.Matrix systems of Painlevé type with scalar coefficients are invariant under conjugations u → T uT −1 , v → T vT −1 by an arbitrary nonsingular matrix T .The corresponding quotient system (that is, the system satisfied by the invariants of this action) does not depend on β, γ.
In Appendix A we provide a miscellaneous information of Hamiltonian non-abelian Painlevé systems including isomonodromoc Lax representations of the form and various links between systems.
In [2,8,3] examples of matrix Hamiltonian P 1 , P 2 , and P 4 systems with non-abelian (but not scalar) coefficients were found.In Section 4 we find P ′ 3 and P 5 systems with one non-abelian parameter.To prove their integrability, we present isomonodromic Lax pairs for them.Appendix B contains explicit formulas for the degenerations into each other of Hamiltonian systems with non-abelian parameter and their Lax pairs.

Non-abelian Hamiltonian ODEs
In this section we define the basic concepts related to non-abelian Hamiltonian systems (see [5,6]).These systems have the form where x 1 , . . ., x N are generators of the free associative algebra A over C. In fact, (4) is the notation for the derivation d z of the algebra A such that d z (x i ) = F i .For any element g ∈ A, the element d z (g) is uniquely determined by the Leibniz rule.Sometimes we use the notation d dz instead of d z .
In the matrix case x i (z) ∈ Mat m (C), the scalar first integrals of systems (4) have the form trace(f (x 1 , ..., x N )).Their generalization to the non-abelian case is the elements of the quotient vector space In this paper we consider the case N = 2 and denote x 1 = u, x 2 = v.It is assumed that the Poisson brackets are generated by the canonical symplectic structure.The Hamiltonian system corresponding to the Hamiltonian H has the form (1), where H ∈ A and ∂ ∂u , ∂ ∂v are non-abelian partial derivatives.Non-abelian derivatives of an arbitrary polynomial f ∈ A are defined by the identity where it is assumed that additional non-abelian symbols du, dv are moved to the right by cyclic permutations of generators in monomials 1 .Notice that in the non-abelian case the partial derivatives are not vector fields.
Performing cyclic permutations in monomials so as to move du, dv to the end of each monomial, we get uvuv du + vuvu du + uvu 2 dv + vu 2 v du + u 2 vu dv.Therefore, Proof.This statement follows from the basic identity for non-abelian partial derivatives, which is true for any g ∈ A (see [5]).
Corollary 1.For any system of the form (1) the elements trace(u 2 v 2 −uvuv) and trace(vu Proof.It is easy to check that and therefore the statement follows from Lemma 1.
1 Such an operation is equivalent to adding a commutator.

Hamiltonian non-abelian Painlevé systems with constant coefficients
For a non-abelian Painlevé system of type i, denote by H i the non-abelian polynomial S (1) i from Assumption 1 that we intend to find.The Hamiltonian for the Painlevé system is defined by the formula By definition, H i should coincide with the polynomials h i defined by ( 2) under the commutative reduction uv = vu.Taking into account Remark 1, it is easy to check that the general ansatz for such non-commutative polynomials can be chosen as follows: The existence of a polynomial S (2) i satisfying Assumption 1 allows one to find the unknown coefficients in these ansatzes.Due to Corollary 1 only the constant α needs to be defined.

Proposition 1. If there exists a polynomial S
(2) i satisfying Assumption 1, then the parameter α in (6) and ( 5) is equal to zero.Other constants remain to be arbitrary.
Proof.Using as an example the case of a non-abelian system P 4 , we will demonstrate how to find the polynomial S (2) 4 .The Hamiltonian of the scalar system P 4 is given by and formula (7) defines its non-abelian generalization S (1) 4 .The polynomial h 2 4 is equal to

Let us construct a general anzats for S
(2) 4 .It is easy to verify that the sets of monomials {vu3 vu, vu2 vu 2 , vu 4 v}, {vu 2 vuv, vuvu 2 v, vuvuvu, vu 3 v 2 }, {vuvuv 2 , vuv 2 uv, vu 2 v 3 } define bases in the vector subspaces of homogeneous polynomials of degrees (4,2), (3,3) and (2,4) in u and v projected onto the quotient space A/[A, A].Therefore, the leading part of the ansatz can be taking as follows: A similar consideration for lower terms leads to is equivalent to an overdetermined algebraic system for the coefficients of H 4 and S (2) 4 .Solving it, we obtain Therefore, the polynomial S (2) 4 has the form The presence of the constants b 1 , d 1 and g 1 is explained by Corollary 1. Similar calculations in the cases of P 5 and P 6 are more laborious.The algebraic system contains the parameter α and it's compatibility conditions give rise to the equality α = 0.
In Appendix A we present isomonodromic Lax representations for the equations of motions defined by the non-abelian Hamiltonians and explicit expressions for limiting transitions connecting Hamiltonian systems.

Systems with non-abelian constants
In the paper [2] the matrix Painlevé-1 equation with an arbitrary constant matrix (or non-abelian constant) h arose.It can be written as the system 3 can be extracted from [8] 4 , while a system of Painlevé-4 type with an arbitrary constant matrix h was found in [3].All these systems are Hamiltonian in the sense of Section 2. The Hamiltonians are given by the formulas Note that after the transition to scalar variables in these systems, the constant h can be reduced to zero by a shift z.In the non-abelian case such a shift is impossible because z is a scalar variable.It is a non-trivial fact that the above three systems possess isomonodromic Lax pairs.There are non-abelian non-Hamiltonian Painlevé systems of P 2 and P 4 type [1,3] with non-abelian coefficients that cannot be constructed using this simple trick.
In this section we present non-abelian Hamiltonian Painlevé systems of P ′ 3 and P 5 type with one arbitrary non-abelian coefficient.In the scalar systems corresponding to them, the parameter h can be reduced to one by a scaling of the variable z.
For the constructed non-abelian systems we were able to find isomonodromic Lax representations only for special values of the parameter β.This is not surprising, since the arbitrariness of the parameters β, γ in non-abelian Hamiltonians is related to their gauge invariance under conjugations, which is absent in the case of systems with non-abelian coefficients.
The scalar Painlevé-3 ′ system depends on two essential parameters while other parameters can be normalized by scallings.In particular, in the case κ 4 = 0 we can reduce κ 4 to 1 by using a scaling of z.After the formal replacement of the scalar parameter κ 4 by h, the polynomial (8) becomes

The corresponding Hamiltonian system has the form
If β = 0 this system has the isomonodromic Lax pair with The same trick produces a P 5 -type Painlevé system with the arbitrary non-abelian constant h.Replacing the parameter κ 4 in ( 6) with h, we get the polynomial which defines the following Hamiltonian system One can check that defines an isomonodromic Lax pair for this system in the case β = −1.One more Hamiltonian system with a non-abelian parameter corresponds to a nonpolynomial degeneration P ′ 3 (D 7 ) of a non-abelian system Painlevé P ′ 3 .The Hamiltonian of this system is given by the formula in the case β = 0 has a Lax pair of the form (21) with matrix coefficients The considered systems with the non-abelian coefficient h and their Lax pairs are related to each other by a limiting transitions given in Appendix B.
System P H 6 is equivalent to the zero-curvature condition (3) with where A.2 Painlevé-5 systems The Hamiltonian system generated by the polynomial ( 6) with α = 0 is written as To get system P H 5 from system P H 6 , one can make the following substitution of variables and parameters: and pass to the limit ε → 0. In order to construct a Lax pair for the system P H 5 , we supplement (12) with the formula λ = 1 + ε z λ.
As a result of the limiting transition, the Lax pair (11) passes to the pair where matrices A 0 , A 1 , A 2 , B 1 , B 0 are given by

Remark 1 .
It is easy to verify that ∂ ∂xi (ab − ba) = 0 for any a, b ∈ A and therefore the nonabelian partial derivatives are well-defined mappings from A to A/[A, A].The Hamiltonian of a non-abelian Hamiltonian system (1) is the equivalence class of the polynomial H ∈ A in A/[A, A] and therefore H is defined up to a linear combination of commutators.Lemma 1.For any system of the form (1) the element I = uv − vu is a non-abelian constant of motion: d z (I) = 0.