**Multi-Poisson approach to the Painlev´** **e** **equations:**

**from the isospectral deformation to the** **isomonodromic deformation**

### Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

### Hayato CHIBA ^{1}

### Apr 26, 2016 **Abstract**

### A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form

*∂X*

*λ*

*/∂t* = [X

*λ*

*, A*

*λ*

### ] in the sense of the isospectral deformation, where *X*

*λ*

*, A*

*λ*

*∈* g depend rationally on the indeterminate *λ* called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation *∂X*

_{λ}*/∂t* = [X

_{λ}*, A*

_{λ}### ] + *∂A*

_{λ}*/∂λ* in the sense of the isomonodromic deforma- tion, which exhibits the Painlev´ e property, is proposed. This method gives a few new Painlev´ e systems of dimension four.

**Keywords: Painlev´** e equations; Lax equations; multi-Poisson structure

**1** **Introduction**

### A diﬀerential equation defined on a complex region is said to have the Painlev´ e property if any movable singularity of any solution is a pole. Painlev´ e and his group classified second order ODEs having the Painlev´ e property and found new six diﬀerential equations called the Painlev´ e equations. Nowadays, it is known that they are written in Hamiltonian forms

### (P

_{J}

### ) : *dq*

*dt* = *∂H*

_{J}*∂p* *,* *dp*

*dt* = *−* *∂H*

_{J}*∂q* *,* *J* = I, *· · ·* *,* VI. (1.1) Among six Painlev´ e equations, the Hamiltonian functions of the first, second and fourth Painlev´ e equations are polynomials in both of the independent variable *t* and the dependent variables (q, p). They are given by

*H*

_{I}

### = 1

### 2 *p*

^{2}

*−* 2q

^{3}

*−* *tq,* (1.2)

*H*

_{II}

### = 1 2 *p*

^{2}

*−* 1

### 2 *q*

^{4}

*−* 1

### 2 *tq*

^{2}

*−* *αq,* (1.3)

*H*

_{IV}

### = *−* *pq*

^{2}

### + *p*

^{2}

*q* *−* 2pqt *−* *αp* + *βq,* (1.4)

1

### E mail address : chiba@imi.kyushu-u.ac.jp

### respectively, where *α, β* *∈* C are arbitrary parameters. Another important property of the Painlev´ e equations is that they are expressed as Lax equations. Let *L*

*λ*

### and *A*

_{λ}### be square matrices which depend rationally on the indeterminate *λ* called the spectral parameter. The Painlev´ e equations are written in Lax form as

*∂L*

*λ*

*∂t* = [L

_{λ}*, A*

_{λ}### ] + *∂A*

*λ*

*∂λ* *,* (1.5)

### for some choice of *L*

_{λ}### and *A*

_{λ}### . This equation arises from the compatibility condition of the two diﬀerential systems

*∂Ψ*

*∂λ* = *L*

_{λ}### Ψ, *∂* Ψ

*∂t* = *A*

_{λ}### Ψ. (1.6)

### Since the monodromy of the former system *∂* Ψ/∂λ = *L*

_{λ}### Ψ is independent of *t* if the equation (1.5) is satisfied, (1.5) is called the isomonodromic deformation equation.

### Another type of the Lax equation is of the form

*∂X*

_{λ}*∂t* = [X

_{λ}*, A*

_{λ}### ], (1.7)

### which is called the isospectral deformation equation because the eigenvalues of the matrix *X*

_{λ}### is independent of *t. There are several systematic ways to construct* isospectral deformation equations [1]. In particular, a Lie algebraic method have been often employed. Let g be a Lie algebra. On the dual space g

^{∗}### , there exists a canonical Poisson structure called the Lie-Poisson structure. If g is equipped with a nondegenerate bilinear symmetric form, the Lie-Poisson structure is also defined on g. Let *P* : *T*

^{∗}### g *→* *T* g be the Poisson tensor and *F* : g *→* C a smooth function.

### Then, the vector field *P dF* on g can be expressed as the Lax equation (1.7) with some *X*

_{λ}*, A*

_{λ}*∈* g [1].

### It is notable that the isospectral deformation equation (1.7) is completely inte- grable for most examples, although the isomonodromic deformation equation (1.5) is not in general; it is believed that solutions of an isomonodromic deformation equation define new functions called the Painlev´ e transcendents.

### In Nakamura [16], a way to obtain the isospectral deformation equation (1.7) from the isomonodromic deformation equation (1.5) by a certain scaling of the time *t* is proposed, which is called the autonomous limit. She proved that the autonomous limits of 6-types of two dimensional Painlev´ e equations and 40-types of four dimen- sional Painlev´ e equations are completely integrable.

### The purpose in the present paper is opposite; a way to construct the isomon- odromic deformation equation (1.5) from the isospectral deformation equation (1.7) will be proposed. Let g be a simple Lie algebra over C . Consider the set of g-valued polynomials of degree *n*

### g

_{n}### := *{* *X*

_{λ}### := *X*

_{0}

*λ*

^{n}### + *X*

_{1}

*λ*

^{n}^{−}^{1}

### + *· · ·* + *X*

_{n}*|* *X*

_{i}*∈* g *}* *,*

### with the indeterminate *λ. This set* g

_{n}### is equipped with a structure of a Lie algebra

### by a certain Lie bracket. At first, the isospectral deformation equation (1.7) on

### g

*n*

### is constructed with the aid of the bi-Poisson theory of Magri et. al [12, 13, 14, 15]. Isospectral deformation equations obtained in this method are shown to be completely integrable (Thm.2.4). Next, we restrict the equations onto a symplectic leaf. Let *φ*

_{1}

*,* *· · ·* *, φ*

_{N}### be Casimir functions of an underlying Poisson structure on g

*n*

### . A symplectic leaf *S* is defined by the level surface of them as

*S* := *{* *φ*

_{i}### = *α*

_{i}### (constant) *|* *i* = 1, *· · ·* *, N* *}* *.*

### Restricted on the leaf *S, the isospectral deformation equation (1.7) becomes an* integrable Hamiltonian system. Since the matrix *X*

*λ*

*∈* g

*n*

### depends on the parameters *α* := (α

_{1}

*,* *· · ·* *, α*

_{N}### ), it is denoted as *X*

_{λ}### = *X*

_{λ}### (t, α).

### Now suppose that there exists a parameter, say *α*

_{j}### , such that the following condition holds

*∂X*

_{λ}*∂α*

_{j}### (t, α) = *∂A*

_{λ}*∂λ* *.* (1.8)

### Eq.(1.7) is put together with Eq.(1.8) to yield

*∂X*

_{λ}*∂t* (t, α) + *∂X*

_{λ}*∂α*

_{j}### (t, α) = [X

_{λ}*, A*

_{λ}### ] + *∂A*

_{λ}*∂λ* *.* Define the Lax matrix *L*

_{λ}### by

*L*

*λ*

### := *X*

*λ*

### (t, α) *|*

*α*

*j*=t

*,*

### where the parameter *α*

_{j}### satisfying the condition (1.8) is replaced by *t. Then, the* above equation is rewritten as the isomonodromic deformation equation (1.5).

### Remark that the isomonodromic deformation equation (1.5) is equivalent to the zero curvature condition of the connection 1 form *L*

_{λ}*dλ* + *A*

_{λ}*dt* on a vector bundle over the (t, λ)-space, while the condition (1.8) is the exactness condition of the connection 1 form *X*

_{λ}*dλ* + *A*

_{λ}*dα*

_{j}### .

### This method is demonstrated for the following three cases (I) g = sl(2, C ), n = 2, (II) g = sl(2, C ), n = 3 and (III) g = so(5, C ), n = 1. For the case (I), the first, second and fourth Painlev´ e equations (1.2), (1.3), (1.4) will be obtained in Section 3.

### More generally, for g = sl(2, C ) with general *n, one can obtain several Painlev´* e hierarchies of dimension 2n *−* 2, including the first Painlev´ e hierarchy (P

_{I}

### )

_{m}### [10, 11, 17], the second-first Painlev´ e hierarchy (P

_{II-1}

### )

_{m}### [5, 6, 10, 11], the second-second Painlev´ e hierarchy (P

_{II-2}

### )

_{m}### and the fourth Painlev´ e hierarchy (P

_{IV}

### )

_{m}### [7, 10]. They are 2m-dimensional Hamiltonian PDEs of the form (m = *n* *−* 1)

###

###

*∂q*

_{j}*∂t*

_{i}### = *∂H*

_{i}*∂p*

_{j}*,* *∂p*

_{j}*∂t*

_{i}### = *−* *∂H*

_{i}*∂q*

_{j}*,* *j* = 1, *· · ·* *, m;* *i* = 1, *· · ·* *, m* *H*

_{i}### = *H*

_{i}### (q

_{1}

*,* *· · ·* *, q*

_{m}*, p*

_{1}

*,* *· · ·* *, p*

_{m}*, t*

_{1}

*,* *· · ·* *, t*

_{m}### )

### (1.9)

### consisting of *m* Hamiltonians *H*

_{1}

*,* *· · ·* *, H*

_{m}### with *m* independent variables *t*

_{1}

*,* *· · ·* *, t*

_{m}### .

### When *m* = 1 (the case (I)), (P

_{I}

### )

_{1}

### and (P

_{IV}

### )

_{1}

### are reduced to the first and fourth

### Painlev´ e equations, respectively. Both of (P

II-1### )

1### and (P

II-2### )

1### coincide with the second Painlev´ e equation, while they are diﬀerent systems for *m* *≥* 2. When *m* = 2 (the case (II)), Hamiltonians of (P

_{I}

### )

_{2}

### , (P

_{II-1}

### )

_{2}

### , (P

_{II-2}

### )

_{2}

### and (P

_{IV}

### )

_{2}

### are given by

### (P

_{I}

### )

_{2}

###

###

*H*

_{1}

### = 2p

_{2}

*p*

_{1}

### + 3p

^{2}

_{2}

*q*

_{1}

### + *q*

_{1}

^{4}

*−* *q*

^{2}

_{1}

*q*

_{2}

*−* *q*

_{2}

^{2}

*−* *t*

_{1}

*q*

_{1}

### + *t*

_{2}

### (q

^{2}

_{1}

*−* *q*

_{2}

### ), *H*

_{2}

### = *p*

^{2}

_{1}

### + 2p

_{2}

*p*

_{1}

*q*

_{1}

*−* *q*

_{1}

^{5}

### + *p*

^{2}

_{2}

*q*

_{2}

### + 3q

_{1}

^{3}

*q*

_{2}

*−* 2q

_{1}

*q*

_{2}

^{2}

### +t

_{1}

### (q

_{1}

^{2}

*−* *q*

_{2}

### ) + *t*

_{2}

### (t

_{2}

*q*

_{1}

### + *q*

_{1}

*q*

_{2}

*−* *p*

^{2}

_{2}

### ),

### (1.10)

### (P

_{II-1}

### )

_{2}

###

###

*H*

_{1}

### = 2p

_{1}

*p*

_{2}

*−* *p*

^{3}

_{2}

*−* *p*

_{1}

*q*

_{1}

^{2}

### + *q*

_{2}

^{2}

*−* *t*

_{1}

*p*

_{2}

### + *t*

_{2}

*p*

_{1}

### + 2αq

_{1}

*,* *H*

_{2}

### = *−* *p*

^{2}

_{1}

### + *p*

_{1}

*p*

^{2}

_{2}

### + *p*

_{1}

*p*

_{2}

*q*

_{1}

^{2}

### + 2p

_{1}

*q*

_{1}

*q*

_{2}

### +t

_{1}

*p*

_{1}

### + *t*

_{2}

### (t

_{2}

*p*

_{1}

*−* *p*

_{1}

*q*

^{2}

_{1}

### + *p*

_{1}

*p*

_{2}

### ) *−* *α(2p*

_{2}

*q*

_{1}

### + 2q

_{2}

### + 2t

_{2}

*q*

_{1}

### ),

### (1.11)

### (P

II-2### )

2###

###

*H*

_{1}

### = *p*

_{1}

*p*

_{2}

*−* *p*

_{1}

*q*

^{2}

_{1}

*−* 2p

_{1}

*q*

_{2}

### + *p*

_{2}

*q*

_{1}

*q*

_{2}

### + *q*

_{1}

*q*

^{2}

_{2}

### + *q*

_{2}

*t*

_{1}

### + *t*

_{2}

### (q

_{1}

*q*

_{2}

*−* *p*

_{1}

### ) + *αq*

_{1}

*,* *H*

_{2}

### = *p*

^{2}

_{1}

*−* *p*

_{1}

*p*

_{2}

*q*

_{1}

### + *p*

^{2}

_{2}

*q*

_{2}

*−* 2p

_{1}

*q*

_{1}

*q*

_{2}

*−* *p*

_{2}

*q*

_{2}

^{2}

### + *q*

_{1}

^{2}

*q*

_{2}

^{2}

### +t

_{1}

### (q

_{1}

*q*

_{2}

*−* *p*

_{1}

### ) *−* *t*

_{2}

### (p

_{1}

*q*

_{1}

### + *q*

_{2}

^{2}

### + *q*

_{2}

*t*

_{2}

### ) + *αp*

_{2}

*,*

### (1.12) (P

IV### )

2###

###

*H*

_{1}

### = *p*

^{2}

_{1}

### + *p*

_{1}

*p*

_{2}

*−* *p*

_{1}

*q*

_{1}

^{2}

### + *p*

_{2}

*q*

_{1}

*q*

_{2}

*−* *p*

_{2}

*q*

^{2}

_{2}

*−* *t*

_{1}

*p*

_{1}

### + *t*

_{2}

*p*

_{2}

*q*

_{2}

### + *αq*

_{2}

### + *βq*

_{1}

*,* *H*

_{2}

### = *p*

_{1}

*p*

_{2}

*q*

_{1}

*−* 2p

_{1}

*p*

_{2}

*q*

_{2}

*−* *p*

^{2}

_{2}

*q*

_{2}

### + *p*

_{2}

*q*

_{1}

*q*

_{2}

^{2}

### +p

_{2}

*q*

_{2}

*t*

_{1}

### + *t*

_{2}

### (p

_{1}

*p*

_{2}

*−* *p*

_{2}

*q*

_{2}

^{2}

### + *p*

_{2}

*q*

_{2}

*t*

_{2}

### ) + (p

_{1}

*−* *q*

_{1}

*q*

_{2}

### + *q*

_{2}

*t*

_{2}

### )α *−* *βp*

_{2}

*,* (1.13) respectively, with arbitrary parameters *α, β* *∈* C . These systems will be obtained from the case (II) g = sl(2, C ), n = 3 in Section 4. In our method, such Hamiltonian PDEs are obtained if there are several Hamiltonian systems written in Lax form (1.7), and if there are several parameters satisfying (1.8); such parameters will be replaced by distinct times *t*

_{1}

*, t*

_{2}

*,* *· · ·* .

### We will find other 4-dimensional Painlev´ e systems with Hamiltonian functions *H*

_{(1,1,2,0)}

### = *−* *p*

^{2}

_{1}

*q*

_{1}

*−* 2p

_{1}

*q*

_{1}

^{2}

### + 2p

_{1}

*q*

_{2}

*−* 2p

_{1}

*p*

_{2}

*q*

_{2}

*−* 2p

_{2}

*q*

_{1}

*q*

_{2}

### +(2p

_{1}

*q*

_{1}

### + 2p

_{2}

*q*

_{2}

### )t + (2α

_{2}

### + 2β

_{2}

### )q

_{1}

### + 2β

_{2}

*p*

_{1}

### + 2β

_{3}

*p*

_{2}

*,* (1.14) *H*

_{(}

_{−}_{1,4,1,2)}

### = *p*

_{1}

*−* *p*

^{2}

_{2}

*−* 2p

_{1}

*q*

_{1}

*q*

_{2}

*−* *p*

_{2}

*q*

^{2}

_{2}

### + 2β

_{3}

*q*

_{2}

### + 2β

_{5}

*q*

_{1}

### + *p*

_{2}

*t,* (1.15)

*H*

_{Cosgrove}

### = *−* 4p

_{1}

*p*

_{2}

*−* 2p

^{2}

_{2}

*q*

_{1}

*−* 73

### 128 *q*

_{1}

^{4}

### + 11

### 8 *q*

^{2}

_{1}

*q*

_{2}

*−* 1 2 *q*

_{2}

^{2}

*−* *q*

1*t* *−* *α*

_{2}

### 48

### (

*q*

1### + *α*

_{2}

### 6

### )

*q*

_{1}

^{2}

*,* (1.16)

### where *α*

_{i}*, β*

_{i}*∈* C are arbitrary parameters (the subscripts for parameters are related to the weighted degrees so that the Hamiltonian functions become quasihomoge- neous, see below). The first two systems will be also obtained from the case (II).

### As far as the author knows, these systems have not appeared in the literature. The last one *H*

_{Cosgrove}

### will be obtained from the case (III) g = so(5, C ), n = 1 in Section 5. If we rewrite the system as a fourth order single equation of *q*

_{1}

### = *y, we obtain*

*y*

^{′′′′}### = 18yy

^{′′}### + 9(y

^{′}### )

^{2}

*−* 24y

^{3}

### + 16t + *αy(y* + 1

### 9 *α).* (1.17)

### This equation was given in Cosgrove [8], denoted by F-VI, without a proof that

### it has the Painlev´ e property. Since this system is obtained as the isomonodromic

### deformation equation in this paper, this equation actually enjoys the Painlev´ e prop- erty. In his paper [8], it is conjectured that this equation defines a new Painlev´ e transcendent (i.e. it is not reduced to known equations). Another expression of the Hamiltonian function of the same system is

*H* e

_{Cosgrove}

### = 2p

_{1}

*p*

_{2}

*−* 18

### 13 *p*

^{2}

_{2}

*q*

_{1}

*−* 2

### 169 *q*

^{4}

_{1}

*−* 180

### 13 *q*

_{1}

^{2}

*q*

_{2}

### + 6q

^{2}

_{2}

*−* 8q

_{1}

*t* + 8

### 9 *α*

_{2}

*q*

_{1}

^{3}

### + 8

### 27 *α*

_{2}

^{2}

*q*

_{1}

^{2}

*.* (1.18) The corresponding Hamiltonian system is also reduced to (1.17).

### Note that all of the Hamiltonian functions above are polynomials in both of the independent variables and the dependent variables. Furthermore, they are semi- quasihomogeneous functions. In general, a polynomial *H(x*

1*,* *· · ·* *, x*

*n*

### ) is called a quasihomogeneous polynomial if there are integers *a*

1*,* *· · ·* *, a*

*n*

### and *h* such that

*H(λ*

^{a}^{1}

*x*

_{1}

*,* *· · ·* *, λ*

^{a}

^{n}*x*

_{n}### ) = *λ*

^{h}*H(x*

_{1}

*,* *· · ·* *, x*

_{n}### ) (1.19) for any *λ* *∈* C . A polynomial *H* is called a semi-quasihomogeneous if *H* is decom- posed into two polynomials as *H* = *H*

^{P}### + *H*

^{N}### , where *H*

^{P}### satisfies (1.19) and *H*

^{N}### satisfies

*H*

^{N}### (λ

^{a}^{1}

*x*

_{1}

*,* *· · ·* *, λ*

^{a}

^{n}*x*

_{n}### ) *∼* *o(λ*

^{h}### ), *|* *λ* *| → ∞* *.*

### The integer wdeg(H) := *h* is called the weighted degree of *H* with respect to the weight wdeg(x

_{1}

*,* *· · ·* *, x*

_{n}### ) := (a

_{1}

*,* *· · ·* *, a*

_{n}### ). For example, if we define degrees of variables by wdeg(q, p, t) = (2, 3, 4) for *H*

_{I}

### , wdeg(q, p, t) = (1, 2, 2) for *H*

_{II}

### and wdeg(q, p, t) = (1, 1, 1) for *H*

_{IV}

### , then Hamiltonian functions have the weighted de- grees 6, 4 and 3, respectively (Table 1). The weights for four dimensional systems above are shown in Table 2. In this paper, these weights are naturally obtained from a suitable definition of weights of entries of a matrix *X*

*λ*

*∈* g

*n*

### and the spectral parameter *λ. In particular, the weights of the Hamiltonian functions are closely* related to the exponents of simple Lie algebras because the Hamiltonian functions are essentially Ad-invariant polynomials of simple Lie algebras. See Chiba [2, 3, 4]

### for the detailed study of the weights of the Painlev´ e equations.

### wdeg(q, p, t) wdeg(H)

### P

_{I}

### (2, 3, 4) 6

### P

_{II}

### (1, 2, 2) 4 P

_{IV}

### (1, 1, 1) 3

### Table 1: Weights for two dimensional Painlev´ e equations.

**2** **Settings**

**2.1** **Lie-Poisson structure on** g _{n}

_{n}

### We define a multi-Poisson structure on a certain Lie algebra following Magri et.

### al [12, 13, 14, 15]. Let (g, [ *·* *,* *·* ]) be a simple Lie algebra over C . Consider the set

### wdeg(q

_{1}

*, p*

_{1}

*, q*

_{2}

*, p*

_{2}

### ) wdeg(t

_{1}

*, t*

_{2}

### ) wdeg(H

_{1}

*, H*

_{2}

### )

### (P

_{I}

### )

_{2}

### (2, 5, 4, 3) 6, 4 8, 10

### (P

_{II-1}

### )

_{2}

### (1, 4, 3, 2) 4, 2 6, 8 (P

_{II-2}

### )

_{2}

### (1, 3, 2, 2) 3, 2 5, 6

### (P

IV### )

2### (1, 2, 1, 2) 2, 1 4, 5

*H*

_{(1,1,2,0)}

### (1, 1, 2, 0) 1 3

*H*

_{(}

_{−}_{1,4,1,2)}

### ( *−* 1, 4, 1, 2) 2 4

*H*

_{Cosgrove}

### (2, 5, 4, 3) 6 8

### Table 2: Weights for four dimensional Painlev´ e equations.

### of g-valued polynomials of degree *n*

### g

_{n}### := *{* *X*

_{λ}### := *X*

_{0}

*λ*

^{n}### + *X*

_{1}

*λ*

^{n}^{−}^{1}

### + *· · ·* + *X*

_{n}*|* *X*

_{i}*∈* g *}* *,* (2.1) with the indeterminate *λ. The bracket defined by*

### [X

_{λ}*, Y*

_{λ}### ]

_{n}### := [X

_{n}*, Y*

_{n}### ] + *λ([X*

_{n}*, Y*

_{n}_{−}_{1}

### ] + [X

_{n}_{−}_{1}

*, Y*

_{n}### ]) + *· · ·* +λ

^{n}### ([X

_{0}

*, Y*

_{n}### ] + [X

_{1}

*, Y*

_{n}_{−}_{1}

### ] + *· · ·* + [X

_{n}*, Y*

_{0}

### ])

### introduces the structure of a Lie algebra on g

_{n}### . Note that [X

_{λ}*, Y*

_{λ}### ]

_{n}### coincides with [X

_{λ}*, Y*

_{λ}### ] expanded in *λ* and truncated at degree *n.*

### It is known that the dual space g

^{∗}### of any Lie algebra g is equipped with a canonical Poisson structure called the Lie-Poisson structure. If a nondegenerate symmetric bilinear form *η* : g *×* g *→* C is defined on g, it induces a Lie-Poisson structure on g. For functions *F, G* : g *→* C , the Poisson bracket on g is defined by *{* *F, G* *}* (X) = *η(X,* [ *∇* *F* (X), *∇* *G(X)]), where* *∇* *F* (X) *∈* g is defined through (dF )

_{X}### (Y ) = *η(* *∇* *F* (X), Y ). To give the Lie-Poisson structure on g

*n*

### , we define a nondegenerate symmetric bilinear form *η* on g

_{n}### by

*η(X*

_{λ}*, Y*

_{λ}### ) :=

### ∑

*n*

*i=0*

### Tr(X

_{i}*Y*

_{n}_{−}_{i}### ),

### by which g

*n*

### is identified with its dual. For a smooth function *F* : g

*n*

*→* C , define the gradient *∇* *F* *∈* g

*n*

### through (dF )(Y

*λ*

### ) = *η(* *∇* *F, Y*

*λ*

### ), and define *∇*

*i*

*F* *∈* g by

*∇* *F* = ( *∇*

*n*

*F* )λ

^{n}### + ( *∇*

*n−*1

*F* )λ

^{n}^{−}^{1}

### + *· · ·* + *∇*

0*F.*

### Using them, the Lie-Poisson bracket on g

_{n}### is given by *{* *F, G* *}*

0 ### := *η(X*

_{λ}*,* [ *∇* *F,* *∇* *G]*

_{n}### )

### = Tr(X

_{0}

*·* [ *∇*

0*F,* *∇*

0*G]) + Tr (X*

_{1}

*·* ([ *∇*

0*F,* *∇*

1*G] + [* *∇*

1*F,* *∇*

0*G])) +* *· · ·* +Tr(X

_{n}*·* ([ *∇*

0*F,* *∇*

*n*

*G] +* *· · ·* + [ *∇*

*n*

*F,* *∇*

0*G]))*

### = *−* Tr( *∇*

0*F* *·* ([X

_{0}

*,* *∇*

0*G] + [X*

_{1}

*,* *∇*

1*G] +* *· · ·* + [X

_{n}*,* *∇*

*n*

*G]))* *− · · ·*

*−* Tr( *∇*

*n−1*

*F* *·* ([X

_{n−1}*,* *∇*

0*G] + [X*

_{n}*,* *∇*

1*G]))* *−* Tr( *∇*

*n*

*F* *·* [X

_{n}*,* *∇*

0*G]).*

### The Poisson tensor (bivector) *P*

0 ### : *T*

^{∗}### g

*n*

*→* *T* g

*n*

### is defined so that *{* *F, G* *}*

0 ### = *dF* (P

_{0}

*dG) =* *η(* *∇* *F, P*

_{0}

*dG) =*

### ∑

*n*

*i=0*

### Tr( *∇*

*i*

*F* *·* (P

_{0}

*dG)*

_{i}### ).

### This implies

*−* (P

0*dG)*

0 ### = [X

0*,* *∇*

0*G] + [X*

1*,* *∇*

1*G] +* *· · ·* + [X

*n*

*,* *∇*

*n*

*G]*

### .. .

*−* (P

_{0}

*dG)*

_{n}_{−}_{1}

### = [X

_{n}_{−}_{1}

*,* *∇*

0*G] + [X*

_{n}*,* *∇*

1*G]*

*−* (P

0*dG)*

*n*

### = [X

*n*

*,* *∇*

0*G].*

### The following expression is useful

*P*

_{0}

### : *dG* *7→ −*

###

###

###

### [X

_{0}

*,* *·* ] [X

_{1}

*,* *·* ] *· · ·* [X

_{n}*,* *·* ]

### .. . . . .

### [X

_{n}_{−}_{1}

*,* *·* ] [X

_{n}*,* *·* ] [X

_{n}*,* *·* ]

###

###

###

###

###

###

*∇*

0*G* .. .

*∇*

*n−*1

*G*

*∇*

*n*

*G*

###

###

###

### =

###

###

###

### [ *∇*

0*G, X*

_{0}

### ] + [ *∇*

1*G, X*

_{1}

### ] + *· · ·* + [ *∇*

*n*

*G, X*

_{n}### ] .. .

### [ *∇*

0*G, X*

_{n}_{−}_{1}

### ] + [ *∇*

1*G, X*

_{n}### ] [ *∇*

0*G, X*

_{n}### ]

###

###

### *.* (2.2)

### It is also represented as a matrix as follows. Let *A* = *A(X) be a representation* matrix of the mapping

*T*

^{∗}### g ( *≃* g) *→* g, *dG* *7→* [X, *∇* *G],* *G* : g *→* C *, X* *∈* g

### with respect to some coordinates on g (here *∇* *G* is the gradient on g). By the definition, *−* *A* is a Poisson tensor of the Lie-Poisson structure on g. Since *A(X) is* linear in *X,* *A(X*

_{λ}### ) is expanded as *A(X*

_{λ}### ) = *λ*

^{n}*A(X*

_{0}

### ) + *λ*

^{n}^{−}^{1}

*A(X*

_{1}

### ) + *· · ·* + *A(X*

_{n}### ).

### Putting *A(X*

_{j}### ) = *A*

_{j}### , *P*

_{0}

### is represented as an (n + 1)dim(g) *×* (n + 1)dim(g) matrix

*P*

_{0}

### = *−*

###

###

###

*A*

_{0}

*· · ·* *A*

_{n}_{−}_{1}

*A*

_{n}*A*

1 *· · ·* *A*

*n*

### .. . . . . *A*

_{n}###

###

### *.* (2.3)

### In what follows, suppose dim(g) = *d,* rank(g) = *h* and let *m*

_{1}

*,* *· · ·* *, m*

_{h}### be ex-

### ponents of g. Let (y

_{1}

*,* *· · ·* *, y*

_{d}### ) be coordinates on g. It is known that the Casimir

### functions of the Lie-Poisson structure on g (i.e. a function *φ* satisfying *{* *F, φ* *}* = 0

### for any *F* : g *→* C ) are the Ad-invariant polynomials denoted by *φ*

_{i}### (y

_{1}

*,* *· · ·* *, y*

_{d}### ), i =

### 1, *· · ·* *, h, and they satisfy deg(φ*

_{i}### ) = *m*

_{i}### + 1.

### Let *x*

*j*

### := (x

*j,1*

*,* *· · ·* *, x*

*j,d*

### ) be coordinates on the *j-th copy of* g (coordinate expres- sion for *X*

*j*

### ) and (x

0*,* *· · ·* *, x*

*n*

### ) coordinates on g

*n*

### . We define the weighted degrees of variables to be

### wdeg(x

_{j}### ) = wdeg(x

_{j,α}### ) = *j,* wdeg(λ) = 1. (2.4) Then, *X*

_{λ}### is quasihomogeneous (homogeneous in the weighted sense) of wdeg(X

_{λ}### ) = *n. Substituting* *y*

*α*

### = *x*

0,α*λ*

^{n}### +x

1,α*λ*

^{n}^{−}^{1}

### + *· · ·* + *x*

*n,α*

### into *φ*

*i*

### (y

1*,* *· · ·* *, y*

*d*

### ) and expanding it in *λ* provide

*φ*

_{i}### (y

_{1}

*,* *· · ·* *, y*

_{d}### ) = *φ*

_{i,0}### (x

_{0}

*,* *· · ·* *, x*

_{n}### )λ

^{(m}

^{i}^{+1)n}

### + *φ*

_{i,1}### (x

_{0}

*,* *· · ·* *, x*

_{n}### )λ

^{(m}

^{i}^{+1)n}

^{−}^{1}

### +

*· · ·* + *φ*

*i,(m*

*i*+1)n

### (x

0*,* *· · ·* *, x*

*n*

### ), *i* = 1, *· · ·* *, h,* which defines polynomials *φ*

_{i,j}### on g

_{n}### satisfying

### deg(φ

_{i,j}### ) = *m*

_{i}### + 1, wdeg(φ

_{i,j}### ) = *j.* (2.5) **Proposition 2.1.**

### (i) *φ*

_{i,j}### depends only on (x

_{0}

*,* *· · ·* *, x*

_{j}### ) for 0 *≤* *j* *≤* *n* *−* 1.

### (ii) *φ*

_{i,j}### (x

_{0}

*, x*

_{1}

*,* *· · ·* *, x*

_{n}### ) = *φ*

_{i,(m}

_{i}_{+1)n}

_{−}_{j}### (x

_{n}*,* *· · ·* *, x*

_{1}

*, x*

_{0}

### ).

### (iii) For each *i, j, α, the derivative* *∂φ*

_{i,j+k}*/∂x*

_{k,α}### is independent of *k* = 0, *· · ·* *, n.*

### (iv) For each *i, j* , the gradient *∇*

*k*

*φ*

_{i,j+k}### is independent of *k* = 0, *· · ·* *, n.*

### (v) For each *i, j, k, the equality*

### ∑

*n*

*l=0*

*A*

_{l}*∂φ*

_{i,j+k}_{−}_{l}*∂x*

_{k}### =

### ∑

*n*

*l=0*

*A*

_{l}*∂φ*

_{i,j}_{−}_{l}*∂x*

_{0}

### = 0 (2.6)

### holds.

### (vi) The Casimir functions of the Lie-Poisson structure *P*

_{0}

### on g

*n*

### are *φ*

_{i,(m}

_{i}_{+1)n}

_{−}_{j}*,* *i* = 1, *· · ·* *, h;* *j* = 0, *· · ·* *, n.*

**Proof.** (i) and (ii) follow from the definition of *φ*

_{i,j}### . (iii) For *y*

*α*

### = ∑

*n*

*k=0*

*λ*

^{n}^{−}^{k}*x*

*k,α*

### , we have

*∂φ*

_{i}*∂y*

_{α}### = *∂x*

_{k,α}*∂y*

_{α}*∂*

*∂x*

_{k,α}(m

### ∑

*i*+1)n

*j=0*

*λ*

^{(m}

^{i}^{+1)n}

^{−}^{j}*φ*

*i,j*

### =

(m

### ∑

*i*+1)n

*j=0*

*λ*

^{m}

^{i}

^{n}^{−}^{j+k}*∂φ*

_{i,j}*∂x*

_{k,α}### =

*m*

### ∑

*i*

*n+k*

*j=k*

*λ*

^{m}

^{i}

^{n}^{−}^{j+k}*∂φ*

_{i,j}*∂x*

_{k,α}*.*

### For the last equality, we used Part (i) combined with Part (ii). Thus we obtain

*∂φ*

*i*

*∂y*

_{α}### =

*m**i**n*

### ∑

*j=0*

*λ*

^{m}

^{i}

^{n−j}*∂φ*

*i,j+k*

*∂x*

_{k,α}*.*

### Since the left hand side is independent of *k, so is each coeﬃcient of* *λ*

^{m}

^{i}

^{n}^{−}^{j}### in the right hand side. Part (iv) immediately follows from (iii).

### (v) The first equality is a consequence of Part (iii). Since *φ*

_{i}### (y) is a Casimir function of the Lie-Poisson structure on g, *Adφ*

_{i}### = 0, where *A* is a matrix defined before. Substituting *y* = ∑

_{n}*k=0*

*λ*

^{n}^{−}^{k}*x*

_{k}### yields 0 = *A* *∂φ*

_{i}*∂y* = (λ

^{n}*A*

0### + *λ*

^{n}^{−}^{1}

*A*

1### + *· · ·* + *A*

*n*

### )

*m**i**n*

### ∑

*j=0*

*λ*

^{m}

^{i}

^{n}^{−}^{j}*∂φ*

_{i,j+k}*∂x*

_{k}### = ∑

*j,l*

*λ*

^{m}

^{i}

^{n}^{−}^{j+n}^{−}^{l}*A*

_{l}*∂φ*

_{i,j+k}*∂x*

_{k}### =

*m*

### ∑

*i*

*n+l*

*j=l*

*λ*

^{m}

^{i}

^{n+n}^{−}^{j}### ∑

*n*

*l=0*

*A*

_{l}*∂φ*

_{i,j+k}_{−}_{l}*∂x*

_{k}*.* This proves the second equality of (v).

### To prove (vi), it is suﬃcient to show

###

###

###

*A*

_{0}

*· · ·* *A*

_{n}_{−}_{1}

*A*

_{n}*A*

_{1}

*· · ·* *A*

_{n}### .. . . . . *A*

_{n}###

###

###

###

###

###

*∂φ*

_{i,(m}

_{i}_{+1)n}

_{−}_{j}*/∂x*

_{0}

*∂φ*

_{i,(m}

_{i}_{+1)n}

_{−}_{j}*/∂x*

_{1}

### .. .

*∂φ*

_{i,(m}

_{i}_{+1)n}

_{−}_{j}*/∂x*

_{n}###

###

### = 0

### for *j* = 0, *· · ·* *, n. This is verified with the aid of Part (v).* □

**Example 2.2.** For g = sl(2, C ), we have *d* = 3, h = 1 and *m*

_{i}### = *m*

_{1}

### = 1. Denote a general element *X*

_{λ}*∈* g

_{n}### as

*X*

_{λ}### = *λ*

^{n}*X*

_{0}

### + *λ*

^{n−1}*X*

_{1}

### + *· · ·* + *X*

_{n}### = *λ*

^{n}### ( *u*

0 *v*

0
*w*

_{0}

*−* *u*

_{0}

### )

### + *λ*

^{n}^{−}^{1}

### ( *u*

1 *v*

1
*w*

_{1}

*−* *u*

_{1}

### )

### + *· · ·* +

### ( *u*

*n*

*v*

*n*

*w*

_{n}*−* *u*

_{n}### )

*.*

### Let (u

*j*

*, v*

*j*

*, w*

*j*

### ) be coordinates on the *j-th copy of* g and (u

0*, v*

0*, w*

0*,* *· · ·* *, u*

*n*

*, v*

*n*

*, w*

*n*

### ) coordinates on g

*n*

### . Then,

*∇*

*j*

*F* =

###

###

### 1 2

*∂F*

*∂u*

_{j}*∂F*

*∂w*

_{j}*∂F*

*∂v*

_{j}*−* 1 2

*∂F*

*∂u*

_{j}###

###

### *,* *A*

_{j}### =

###

### 0 *v*

_{j}*−* *w*

_{j}*−* *v*

_{j}### 0 2u

_{j}*w*

_{j}*−* 2u

_{j}### 0

###

### *.*

### The Casimir function on g is given by *φ*

_{i}### = *φ* = *u*

^{2}

### + *vw. Then, the functions* *φ*

_{i,j}### = *φ*

_{j}### are defined by expanding

### (λ

^{n}*u*

_{0}

### + *· · ·* + *u*

_{n}### )

^{2}

### + (λ

^{n}*v*

_{0}

### + *· · ·* + *v*

_{n}### )(λ

^{n}*w*

_{0}

### + *· · ·* + *w*

_{n}### ) in *λ. This gives*

*φ*

_{j}### = ∑

*k+l=j*

### (u

_{k}*u*

_{l}### + *v*

_{k}*w*

_{l}### ) *,* *j* = 0, *· · ·* *,* 2n.

### Note that they are coeﬃcients of *−* det *X*

*λ*

### . The Casimir functions of g

*n*

### are given by *φ*

*j*

### for *j* = *n,* *· · ·* *,* 2n.

**2.2** **Multi-Poisson structure on** g ^{0} _{n}

_{n}

### In general, a manifold *M* is called a bi-Poisson manifold if **(i)** there are two Poisson brackets *{* *,* *}*

0 ### and *{* *,* *}*

1### , and

**(ii)** the linear combination *{* *,* *}*

0### + *t* *{* *,* *}*

1 ### is also a Poisson bracket for any *t* *∈* C .

### See [12, 13, 14, 15] for applications of bi-Poisson manifolds to integrable systems.

### Here, we introduce a bi-Poisson structure on g

_{n}### following [13]. The shift operator *X*

*λ*

*7→* *X*

*λ+t*

### defines an automorphism of g

*n*

### with a parameter *t* *∈* C . It induces a deformation, denoted by *{* *,* *}*

*t*

### , of the Lie-Poisson bracket *{* *,* *}*

0### . Let

*{* *,* *}*

*t*

### = *{* *,* *}*

0### + *t* *{* *,* *}*

1### + *· · ·* + *t*

^{n+1}*{* *,* *}*

*n+1*

### + *· · ·*

### be its expansion. Magnano and Magri [13] proved that each *{* *,* *}*

*i*

### (i = 0, *· · ·* *, n+*

### 1) and their any linear combination satisfy the axiom of a Poisson bracket. Hence, g

_{n}### has *n* + 2 compatible Poisson brackets and it becomes a multi-Poisson manifold.

### Their Poisson tensors are

*P*

_{1}

### =

###

###

###

###

### 0 0 0 *· · ·* 0

### 0 *−* *A*

_{1}

*−* *A*

_{2}

*· · · −* *A*

_{n}### .. . .. . .. . . . .

### 0 *−* *A*

_{n}_{−}_{1}

*−* *A*

_{n}### 0 *−* *A*

_{n}###

###

###

### *,*

*P*

_{k+1}### =

###

###

###

###

###

###

###

### 0 0 *· · ·* 0 0 *· · ·* 0

### 0 *A*

0
### .. . . . . .. . 0 *A*

_{0}

*· · ·* *A*

_{k}_{−}_{1}

### 0 *−* *A*

_{k+1}*· · · −* *A*

_{n}### .. . .. . . . .

### 0 *−* *A*

_{n}###

###

###

###

###

###

###

*,* *k* = 1, *· · ·* *, n* *−* 1,

*P*

_{n+1}### =

###

###

###

###

### 0 0 *· · ·* 0 0

### 0 *A*

_{0}

### 0 *A*

_{0}

*A*

_{1}

### .. . . . . .. . .. . 0 *A*

0 *· · ·* *A*

*n−*2

*A*

*n−*1

###

###

###

### *.*

### (P

_{0}

### is the same as before). Let g

^{0}

_{n}### be a submanifold of g

_{n}### defined by *x*

_{0}

### = constant;

### g

^{0}

_{n}### := *{* *X*

*λ*

### = *X*

0*λ*

^{n}### + *X*

1*λ*

^{n}^{−}^{1}

### + *· · ·* + *X*

*n*

*|* *X*

0 ### = constant *} ⊂* g

*n*

*.* (2.7)

### Since the first row and column of *P*

1*,* *· · ·* *, P*

*n+1*

### are zero (i.e. *x*

0 ### = (x

0,1*,* *· · ·* *, x*

0,d### ) are Casimir functions of them), the restrictions of them on g

^{0}

_{n}### define a multi-Poisson structure on g

^{0}

_{n}### , whose brackets and tensors are again denoted by ( *{* *,* *}*

*i*

*, P*

_{i}### ).

### The tensors are given by

*P*

_{1}

### =

###

###

###

*−* *A*

_{1}

*−* *A*

_{2}

*· · · −* *A*

_{n}### .. . .. . . . .

*−* *A*

_{n}_{−}_{1}

*−* *A*

_{n}*−* *A*

_{n}###

###

### *,*

*P*

_{k+1}### =

###

###

###

###

###

###

*A*

_{0}

### . . . .. . *A*

_{0}

*· · ·* *A*

_{k}_{−}_{1}

*−* *A*

_{k+1}*· · · −* *A*

_{n}### .. . . . .

*−* *A*

_{n}###

###

###

###

###

###

*,* *k* = 1, *· · ·* *, n* *−* 1,

*P*

_{n+1}### =

###

###

###

*A*

_{0}

*A*

_{0}

*A*

_{1}

### . . . .. . .. . *A*

_{0}

*· · ·* *A*

_{n}_{−}_{2}

*A*

_{n}_{−}_{1}

###

###

### *.*

### For *i* = 1, *· · ·* *, h* and *j* = 1, *· · ·* *,* (m

*i*

### + 1)n, define functions *ψ*

*i,j*

### on g

^{0}

_{n}### by *ψ*

_{i,j}### (x

_{1}

*,* *· · ·* *, x*

_{n}### ) := *φ*

_{i,j}*|*

g^{0}

_{n}### = *φ*

_{i,j}*|*

*x*0=constant

*.*

### (we do not define *ψ*

_{i,0}### because *φ*

_{i,0}### is constant on g

^{0}

_{n}### ).

**Proposition 2.3.**

### (i) Casimir functions of *P*

*k+1*

### are *ψ*

*i,j*

### (i = 1, *· · ·* *, h) for* *j* = 1, 2, *· · ·* *, k* and for *j* = *m*

*i*

*n* + *k* + 1, m

*i*

*n* + *k* + 2, *· · ·* *,* (m

*i*

### + 1)n.

### (ii) Casimir functions of the combination *λP*

_{k+1}*−* *P*

_{k}### are *ψ*

_{i,j}### (i = 1, *· · ·* *, h) for* *j* = 1, 2, *· · ·* *, k* *−* 1 and for *j* = *m*

_{i}*n* + *k* + 1, m

_{i}*n* + *k* + 2, *· · ·* *,* (m

_{i}### + 1)n, and

*λ*

^{m}

^{i}

^{n}*ψ*

_{i,k}### + *λ*

^{m}

^{i}

^{n}^{−}^{1}

*ψ*

_{i,k+1}### + *· · ·* + *ψ*

_{i,m}

_{i}

_{n+k}*,* (i = 1, *· · ·* *, h).*

### (iii) Let *F* : g

^{0}

_{n}*→* C be a smooth function. The diﬀerential equation for the vector field (λP

_{k+1}*−* *P*

_{k}### )dF is expressed in Lax form as

*d*

*dt* *X*

_{λ}### = [X

_{λ}*,* *∇*

*k*

*F* ], *X*

_{λ}### = *λ*

^{n}*X*

_{0}

### + *λ*

^{n}^{−}^{1}

*X*

_{1}

### + *· · ·* + *X*

_{n}*.* (iv) Define the function *G*

_{i,k,j}### to be

*G*

*i,k,j*

### = *−* (

*λ*

^{j}^{−}^{1}

*ψ*

*i,k*

### + *λ*

^{j}^{−}^{2}

*ψ*

*i,k+1*

### + *· · ·* + *ψ*

*i,k+j−*1

### ) *.*

### Then, the equality

*P*

_{k+1}*dψ*

_{i,k+j}### = *P*

_{k}*dψ*

_{i,k+j}_{−}_{1}

### = (λP

_{k+1}*−* *P*

_{k}### )dG

_{i,k,j}### holds for *i* = 1, *· · ·* *, h, j* = 1, *· · ·* *, m*

*i*

*n* and *k* = 1, *· · ·* *, n. In particular, the vector* field *P*

_{k+1}*dψ*

_{i,k+j}### is independent of *k* and the equation for it is expressed in Lax form as

*d*

*dt* *X*

_{λ}### = [X

_{λ}*,* *∇*

*k*

*G*

_{i,k,j}### ].

### (v) The vector fields *P*

_{k+1}*dψ*

_{i,k+j}### for *i* = 1, *· · ·* *, h* and *j* = 1, *· · ·* *, m*

_{i}*n* commute with each other (note that it is zero when *j /* *∈ {* 1, *· · ·* *, m*

*i*

*n* *}* ).

**Proof.** (i) and (ii) can be verified by a straightforward calculation with the aid of Prop.2.1 (v). To prove (iii), note that the vector field *P*

_{k+1}*dF* is written as

*P*

_{k+1}*dF* =

###

###

###

###

###

###

### [X

0*,* *·* ] . . . .. . [X

_{0}

*,* *·* ] *· · ·* [X

_{k}_{−}_{1}

*,* *·* ]

*−* [X

_{k+1}*,* *·* ] *· · · −* [X

_{n}*,* *·* ] .. . . . .

*−* [X

_{n}*,* *·* ]

###

###

###

###

###

###

###

###

###

###

###

*∇*

1*F* .. .

*∇*

*k*

*F*

*∇*

*k+1*

*F* .. .

*∇*

*n*

*F*

###

###

###

###

### *,*

### and similarly for *P*

*k*

*dF* . Using them, write down the equation of *X*

*j*

### for the vector field (λP

_{k+1}*−* *P*

_{k}### )dF . For example, the equation for *X*

_{1}

### is *dX*

_{1}

*/dt* = *λ[X*

_{0}

*,* *∇*

*k*

*F* ] *−* [X

_{0}

*,* *∇*

*k−*1

*F* ]. Summing up the equations of *λ*

^{n}^{−}^{j}*X*

_{j}### proves the desired result.

### (iv) Since *λ*

^{m}

^{i}

^{n}*ψ*

_{i,k}### + *· · ·* + *ψ*

_{i,m}

_{i}

_{n+k}### is the Casimir of *λP*

_{k+1}*−* *P*

_{k}### , we have (λP

_{k+1}*−* *P*

_{k}### )d(λ

^{m}

^{i}

^{n}*ψ*

_{i,k}### + *λ*

^{m}

^{i}

^{n−1}*ψ*

_{i,k+1}### + *· · ·* + *ψ*

_{i,m}

_{i}

_{n+k}### ) = 0.

### Expanding this yields the first equality. The second equality is confirmed by a straightforward calculation.

### (v) Due to Part (iv), we can assume that *k* = *n. Because of the property* [P

_{n+1}*dF, P*

_{n+1}*dG] =* *P*

_{n+1}*d* *{* *G, F* *}* of a Poisson bracket (the left hand side is the Lie bracket for vector fields), it is suﬃcient to show the equality *{* *ψ*

_{i}*′*

*,j*

^{′}*, ψ*

_{i,j}*}*

*n+1*

### = 0 for *i, i*

^{′}### = 1, *· · ·* *, h* and *j, j*

^{′}### = 1, *· · ·* *,* (m

_{i}### + 1)n. When *j* = 1, *· · ·* *, n, it is trivial because* *ψ*

_{i,j}### is the Casimir of *P*

_{n+1}### . Next, we have

*{* *λ*

^{m}

^{i}

^{n}*ψ*

_{i}*′*

*,k*

### + *λ*

^{m}

^{i}

^{n}^{−}^{1}

*ψ*

_{i}*′*

*,k+1*

### + *· · ·* + *ψ*

_{i}*′*

*,m*

*i*

*n+k*

*, ψ*

_{i,n+j}*}*

*n+1*

### = *⟨* *d(λ*

^{m}

^{i}

^{n}*ψ*

*i*

^{′}*,k*

### + *λ*

^{m}

^{i}

^{n}^{−}^{1}

*ψ*

*i*

^{′}*,k+1*

### + *· · ·* + *ψ*

*i*

^{′}*,m*

*i*

*n+k*

### ) *, P*

*n+1*

*dψ*

*i,n+j*

*⟩*

### = *⟨* *d(λ*

^{m}

^{i}

^{n}*ψ*

_{i}*′*

*,k*

### + *λ*

^{m}

^{i}

^{n}^{−}^{1}

*ψ*

_{i}*′*

*,k+1*

### + *· · ·* + *ψ*

_{i}*′*

*,m*

*i*

*n+k*

### ) *, P*

_{k+1}*dψ*

_{i,k+j}*⟩*

### = *⟨* *d(λ*

^{m}

^{i}

^{n}*ψ*

_{i}*′*

*,k*

### + *λ*

^{m}

^{i}

^{n}^{−}^{1}

*ψ*

_{i}*′*

*,k+1*

### + *· · ·* + *ψ*

_{i}*′*

*,m*

*i*

*n+k*

### ) *,* (λP

_{k+1}*−* *P*

_{k}### )dG

_{i,k,j}*⟩*

### = *−⟨* *dG*

_{i,k,j}*,* (λP

_{k+1}*−* *P*

_{k}### )d(λ

^{m}

^{i}

^{n}*ψ*

_{i}*′*

*,k*

### + *λ*

^{m}

^{i}

^{n−1}*ψ*

_{i}*′*

*,k+1*

### + *· · ·* + *ψ*

_{i}*′*

*,m*

*i*

*n+k*

### ) *⟩* = 0.

### This provides

*{* *ψ*

_{i}*′*

*,k*

*, ψ*

_{i,n+j}*}*

*n+1*

### = *· · ·* = *{* *ψ*

_{i}*′*

*,m*

*i*

*n+k*

*, ψ*

_{i,n+j}*}*

*n+1*

### = 0,

### for any *k* = 1, *· · ·* *, n* and any *j* = 1, *· · ·* *, m*

*i*

*n, which completes the proof.* □

**Theorem 2.4.** Suppose that the constant *x*

_{0}

### for the definition of g

^{0}

_{n}### is chosen so that the functions *{* *ψ*

_{i,j}*}*

*i,j*

### are functionally independent. Then, the vector field *P*

_{k+1}*dψ*

_{i,k+j}### , which is independent of *k, is completely integrable in the Liouville sense* for any *i* and *j.*

**Proof.** Recall dim(g) = *d* and rank(g) = *h. Thus, dim(g*

^{0}

_{n}### ) = *nd. Since* *P*

_{k+1}### has *nh* Casimir functions, the dimension of a symplectic leaf *S* of *P*

_{k+1}### is *n(d* *−* *h).*

### On the leaf *S, the vector fields* *{* *P*

_{k+1}*dψ*

_{i,k+j}*}*

*i,j*

### define *n(d* *−* *h)-dim Hamiltonian* systems, among which nonzero vector fields are for *i* = 1, *· · ·* *, h* and *j* = 1, *· · ·* *, m*

_{i}*n.*

### Further, these nonzero vector fields commute with each other and they are linearly independent due to the assumption. The number of the nonzero vector fields is

### ∑

*h*

*i=1*

*m*

*i*

*n* = 1

### 2 (dim(g) *−* rank(g))n = 1

### 2 *n(d* *−* *h) =* 1

### 2 dim(S).

### Hence, the Liouville theorem shows that the vector fields are integrable. □

### In what follows, we suppose the above assumption; the constant *x*

_{0}

### for the def- inition of g

^{0}

_{n}### is chosen so that the functions *{* *ψ*

_{i,j}*}*

*i,j*

### are functionally independent.

### That is, the diﬀerentials *{* *dψ*

_{i,j}*}*

*i,j*

### are linearly independent except for finite points.

**2.3** **Symplectic reduction**

### The next purpose is to perform a symplectic reduction [12, 13, 14, 15].

**Lemma 2.5.** The *h* dimensional distribution *D* defined by *D* = span *{* *P*

*k*

*dψ*

*i,k*

*|* *i* = 1, *· · ·* *, h* *}*

### is integrable in the Frobenius sense. The vector fields *P*

_{k}*dψ*

_{i,k}### are linear for *i* = 1, *· · ·* *, h.*

**Proof.** The first statement follows from Prop.2.3(v). Since *P*

_{k}*dψ*

_{i,k}### is independent of *k, we obtain* *P*

_{k}*dψ*

_{i,k}### = *P*

_{1}

*dψ*

_{i,1}### . Since wdeg(ψ

_{i,1}### ) = 1, *dψ*

_{1,i}

### is a constant, while *P*

_{1}

### is linear in (x

_{1}

*,* *· · ·* *, x*

_{n}### ). □

### The diﬀerential equation for *P*

_{k}*dψ*

_{i,k}### = *P*

_{1}

*dψ*

_{i,1}### is given by *d*

*dt* *X*

_{λ}### = [X

_{λ}*,* *∇*

1*G*

_{i,1,1}### ] = [ *∇*

1*ψ*

_{i,1}*, X*

_{λ}### ].

### Since *∇*

1*ψ*

_{i,1}### is independent of *λ, this is decomposed as* *d*

*dt* *X*

_{k}### = [ *∇*

1*ψ*

_{i,1}*, X*

_{k}### ], *k* = 1, *· · ·* *, n.*

### In coordinates, it is expressed as *dx*

*k*

*dt* = *−* *A*

*k*

*∂ψ*

*i,1*

*∂x*

_{1}