数理解析研究所講究録
931 巻 1995 年 20-33

20

Holonomic deformation of
linear differential equations ofthetype and polynomial
Hamiltonian structure$A_{3}$

(劉

勅夫 )

(岡本

徳、朗)

OKMOTO KAZUO &LIU DEMING
Department of Mathematical$\mathrm{S}\mathrm{c}\mathrm{i}\mathrm{e}\tilde{\mathrm{n}}$ces University of Tokyo

Abstract

We study the theory of the holonomic deformation for the linear
differential equation of thetype and show that the holonomic deformation of the equation is represented by the Hamiltonian system withthe polynomial Hamiltonian. We also give the particular solutions of
the polynomial Hamiltonian system.
$A_{3}$

Introduction.

$0$

In this paper, we consider following linear differential equation with irregular singular point, $x.=\infty$ and three non logarithmic singular points, $x=$,$\lambda_{1},$$\lambda_{2},$(0.1)$\lambda_{3}$$\frac{d^{2}y}{dx^{2}}+p_{1}(X)\frac{dy}{dx}+p_{2}(x)y=0$

defined on the Riemann scheme$\mathrm{P}^{1}$,, such that$p_{1}(x)=-2x4- \sum_{)\mathrm{t}j}j_{S_{j}}x^{j-1}-\frac{3}{4}s^{2}3^{-\sum_{k)}\frac{1}{x-\lambda_{k}’}}\mathrm{t}$(0.2)$p_{2}(x)=-(2 \alpha+1)x^{3}-2\sum_{(j)}HjX3-j+\sum_{\langle k)}\frac{\mu_{k}}{x-\lambda_{k}}$.

We suppose that $2\alpha+1$ is not an integer through out this paper, Thisequation has an irregularity at $x=\infty$ of the Poincar\’e rank 5, and three regular regular points $x=\lambda_{k}(k=1,2,3)$ . We make the following assumption:

21

None of $x=\lambda_{k}(k=1,2,3)$ is logarithmic singularity.
The aim of this paper is to study the holonomic deformation of $(0.1)-(0.2)$
under the assumption (A). The main results of this paper are as follows:(A)the holonomic deformaUnder the assumption
Main theorem.
tion of the equation (0.4) is governed by the completely integrable Hamiltoniansystem of partial differential equations:$(A)_{J}$$\frac{\partial\lambda_{k}}{\partial s_{j}}=\frac{\partial H_{j}}{\partial\mu_{k}}$$(k,j=1,2,3)$ .

$(H)$$\frac{\partial\mu_{k}}{\partial s_{j}}=-\frac{\partial H_{j}}{\partial\lambda_{k}}$The linear equation $(0.1)_{-}(0.2)$ is a particular case of an equation of theform (0.1), such that$p_{1}(_{X})=-2x- \sum_{j=1}g+1jtjx^{j}-1-\sum^{g}gk=1\frac{1}{x-\lambda_{k}’}$

(0.3)

.

$p_{2}(x)=-(2 \alpha+1)x^{g}-2\sum_{=j1}Hgj^{X}\mathit{9}-j+\sum_{k=1}^{g}\frac{\mu_{k}}{x-\lambda_{k}}$with the following Riemann scheme:(0.4)$(x= \lambda_{k}02\frac{x=\infty}{\frac{02}{g+2}0t00_{\mathit{9}g-}t1t1-\alpha+\frac{1}{2}\mathrm{o}\cdot.\cdot \mathrm{o}\alpha+\frac{1}{2}}.\cdot\cdot)$$(k=1, \cdots,g)$means that, at the irregular point $x=\infty$ , theHere the symbol in$(0.\dot{1})-(0.2)$
solutions of the form:a system ofadmitsequation$(0.4_{\mathrm{g}})$form.a.1$y_{1}..=x^{-\frac{1+2\alpha}{2}\sum_{\geq}hX^{-}}n01,nn$

,
4 ‘

(05)
$y_{2}=x^{\frac{2\alpha-1}{2}} \exp\{\frac{2}{5}x^{5}+\sum_{\mathrm{t}^{i})} six^{i}+\frac{3}{4}S_{3^{2}}X\}$$\sum_{n\geq 0}h_{2},Xn-n$.

Note that the Poincar\’e rank at $x=\infty$ of the linear equation $(0.1)-(0.3)$. The principal parts of the formal solutions are given by the primitiveis $g+2$

22function of the polynomial representing the versal deformation of the simple
type, so we call the linear equation $(0.1)-(0.3)$ as the
singularity of theequation oftype.
When considering the holonomic deformation of equation of thetype,
we obtain the Hamiltonian structure$A_{g}$$A_{g}$$A_{1}$$(\lambda_{1}, \mu_{1}, H_{1,1}t)$.

which determines the Hamiltonian system, equivalent to the second Painlev\’eequation, see [1].
When $g=2$ , it is known [2] that the holonomic deformation is governedby the Hamiltonian system with respect to the canonical variables:$(\lambda_{1}, \lambda_{2,\mu_{1},\mu}2, H1, H2,t1,t2)$.

On the other hand, in the case $g\geq 3$ , the quantities $H=(H_{1}, \cdots, H_{g})$ anddo not compose the Hamiltonian structure. In fact, even in
the case $g=3,$
have determine the variables $s=(\mathit{8}2, s2,s3)$ such that$t=(t_{1}, \cdot, .,t_{g})$$\mathrm{w}\dot{\mathrm{e}}$(0.6)$\mathrm{t}\dot{\mathrm{o}}$$s_{1}=t_{1}+ \frac{3}{4}t_{3}^{2}$,$s_{2}=t_{2}$,$s_{3}=t_{3}$,and then obtain the Hamiltonian structure.

$(\lambda_{1}, \lambda_{2}, \lambda_{3},\mu_{1},\mu 2, \mu 3, H1, H2, H_{3,\mathit{8}}1, s_{2,3}s)$

We are studying in the present paper the holonomic deformation of equatype. As to the case of $g\geq 4$ , we will study in the future.
tion of the$A_{3}$

1

Holonomic deformation of linearof the second order.

In this section, we recall the theory of thedifferential equation of the form:(1.1)$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{;}\mathrm{o}\mathrm{n}$omic deformation of linear$\mathrm{h}\mathrm{o}1_{0}.\mathrm{n}$$\frac{d^{2}y}{dx^{2}}+p_{1}(X, s)\frac{dy}{dx}+p2(X, \mathit{8})y=0$

viewing $s=(s_{1g}, \cdots, s)$ as the deformation variables.

,

23

the equation (1.1) has a fundamental system of solutions whose monodromy and Stokes multipliers are independent of the pa$A_{j}(x),$ $B_{j(x})$ ,rameter , if and only if there exist rational functions ofsuch that the following system of partial differential equations is completely
integrable.

Proposition 1.1.

$x,$

$s$

$\frac{\partial^{2}y}{\partial x^{2}}+p_{1}(_{X}, s)\frac{\partial y}{\partial x}+p2(x, s)y=0$

(1.2)

.

$\frac{\partial y}{\partial_{\mathit{8}_{j}}}=B_{j}(x)y+Aj(_{X)\frac{\partial y}{\partial x}}$The conditions of the complete integrability of (1.2)

Proposition 1.2.
are given by:
(1.3)

(1.4)

,

( $i,j=1$ , cdots, ).
$g$$\frac{\partial A_{j}}{\partial s_{i}}+A_{j^{\frac{\partial A_{i}}{\partial x}}}=\frac{\partial A_{i}}{\partial s_{j}}+A_{i^{\frac{\partial A_{j}}{\partial x}}}$.

$A_{\mathrm{j}} \frac{\partial^{3}A_{j}}{\partial x^{3}}-2\frac{\partial}{\partial x}(A^{2}jP)+2A_{j}\frac{\partial P}{\partial t_{j}}=0$$(j=1, *.\cdot\cdot.

,g)$

.

where

(1.5.).

$P(x, s)=-p2(x,s)+ \frac{1}{4}p_{1}(2SX,)+\frac{1}{2}\frac{\partial}{\partial x}p1(_{X,\mathit{8}})$If we make the change of the unknown function:$y–\Phi(x)_{Z}$

(1.6)

,

$\Phi(x)=\exp(-\frac{1}{2}\int^{x}p_{1}(X, t)dx)$.

then (1.1) is transformed to an equation of the form:(1.7)$\frac{d^{2}z}{dx^{2}}=P(x, s)z$

Proposition 1.3.

$\cdot$.

The holonomic deformation of (1.1) is reduced tothat of (1.7).
A linear equation of the form (1.7) is called the -type equation.
Finally the holonomic deformation of (1.1) is reduced to the existence of, satisfying the system $(1.3)-(1.4)$ of partial differrational functions
ential equations. We will call (1.2) as the extended system of (1.1) and theas the deformation functions.
functions$\mathrm{S}\mathrm{L}$$A_{j}(x)$$A_{j}(x)$$)\mathrm{y}$242Deformation function$A_{j}(x)$.

In the following of this paper, we consider the holonomic deformation of
linear equations of the form:
(2.1)

,

$\frac{d^{2}y}{dx^{2}}+p_{1}(_{X}, s)\frac{dy}{dx}+p2(X, S)y=0$$p_{1}(x)=-2X4- \sum j\mathit{8}_{j3}\{j)X^{j}-1-\frac{3}{4}s^{2}-\sum_{\mathrm{t}k)}\frac{1}{x-\lambda_{k}}$

,

(2.2)
$p_{2}(x)=-(2 \alpha+1)x^{3}-2\sum_{\mathrm{t}j)}H_{j}x+\sum_{k)}\mathrm{s}_{-}j\frac{\mu_{k}}{x-\lambda_{k}’}\mathrm{t}$under the assumption (A). For the limiting pages, here we only can give theresults, omit their proofs.
Firstly we determine the deformation functions.

Proposition 2.1.
(2.3)

The function $A_{j}(x)(j=1,2,3)$ are given as follows:$A_{j}(x)= \frac{Q_{j-1}(x)}{\Lambda(x)}$

,

where
$Q_{j-1}(x)= \sum_{=}j-1i0q^{(}i(j))_{X^{j1-}}-iS$,$\Lambda(x)=\prod(_{X}-\lambda_{j}j=13)$.

The explicit form of $q_{i}^{\langle j)}.(s.).(ja=1,2,3;i=0, \cdots ,j-1)$ will be given in
Section 4
In order to prove this proposition, we shall first study theof$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}.\mathrm{t}\mathrm{y}$$A_{j}(x)$.

Lemma 2.1.

For any fixed , each $A_{j}(x)$ is holomorphic on

Lemma 2.2.
Lemma 2.3.

For $k=1,2,3,$
is a pole of the first order of $A_{j}(x)$ .
$A_{i}(x)$ admits a zeroof order $4-j$ at $x=\infty$ .

$s$$x=\lambda_{k}$$\mathrm{C}\backslash \{\lambda_{1,2}\lambda, \lambda_{3}\}$.

25

Equations of the SL-type.

3

In this section, we will investigate the equation of the SL-type:$\frac{d^{2}z}{dx^{2}}=P(_{X,S})Z$

(3.1)

,

$P(x,s)=-p2(x,s)+ \frac{1}{4}p_{1}2(_{X,s})+\frac{1}{2}\frac{\partial}{\partial x}p1(X, S)$.

$P(x, s)$ can be written in the following form:$P(x, s)=x^{8}+x^{3} \sum_{j=0}^{3}F_{j^{X}}j+2\sum_{j=1}I\mathrm{f}\mathrm{j}x^{3-j}3$(3.2)$- \sum_{1k=}^{3}\frac{\nu_{k}}{x-\lambda_{k}}+\frac{3}{4}\sum_{k=1}^{3}\frac{1}{(x-\lambda_{k})^{2}}$,where, we have following relations:$F_{3}=3s_{3}$,$F_{2}=2s_{2}$,$F_{1}=s_{1}+3s_{3}^{2}$,$F_{0}=2\alpha+3s_{2}s_{3}$.

,$I \mathrm{f}_{1}=H_{1}+\frac{3}{4}s_{3}(_{S}1+\frac{3}{4}S_{3}2)+\frac{1}{2}s^{2}2^{+\frac{1}{2}}k\sum^{3}=1\lambda_{k}$(3.3)$K_{2}=H_{2}+ \frac{3}{4}s_{3}+\frac{1}{2}s_{2}(S_{1}+\frac{3}{4}s_{3}^{2})+\frac{1}{2}.\sum_{k=1}^{3}\lambda_{k}^{2}$,$I \mathrm{f}_{3}=H3+\mathit{8}2+\frac{1}{8}(S1+\frac{3}{4}s32)^{2}+\frac{1}{2}\sum\lambda 3+k=13\mathit{8}_{3}k\frac{3}{4}\dot{\sum^{3}}k=1\lambda_{k}$(3.4)$\nu_{k}=\mu_{k}-\frac{1}{2}l=1\sum_{\neq;k}^{3}\frac{1}{\lambda_{k}-\lambda_{l}}-\frac{1}{2}\sum_{i=1}^{3}iSi\lambda_{k^{-}3^{-}}^{i1}-\frac{3}{8}S2\lambda_{k}^{4}$.
.

For the simplicity of presentation, we put:
(3.5)
$N_{k}= \frac{1}{\Pi_{i=1;\neq}^{3}k}(\lambda_{k}-\lambda_{i})$where$\sigma_{j}(\check{\lambda}_{k})$,$N^{jk}=(-1)^{j-1}\sigma_{j-1}(\check{\lambda}_{k})$,$\Lambda(x)=\prod(_{X}-\lambda_{i}i=13)$,

denotes the j-th elementary symmetric polynomials of two

26

variables, $\lambda_{j}(j=1,2,3;\neq k)$ .
Proposition 3.1.
The Hamiltonian functions $H_{j}(j=1,2,3)$ are given by(3.6)$H_{j}= \frac{1}{2}\sum_{\mathrm{t}^{k})}[NkNjk\mu k^{-}U2jk\mu_{k^{-N}}kN^{jk}(2\alpha+1)\lambda_{k}^{3}]$

,

where
$U_{jk}=N_{k}N^{jk}(2 \lambda_{k^{+S_{3}\lambda_{k}^{2}+S}}^{4}322\lambda k+s_{1}+\frac{3}{4}S_{3}^{2})-\iota=\sum_{1;\neq k}^{3}\frac{N_{k}N^{jk}+N\iota^{N}jl}{\lambda_{l}-\lambda_{k}}$Proposition 3.2.
written follows:.

In the linear equation $(3.1)-(3.2),$ $K_{j}(j=1,2,3)$ are$a\mathit{8}$

(3.7)

,

$IC_{\mathrm{j}}= \frac{1}{2}\sum_{=k1}^{3}(NkNjk2\nu_{k^{-\sum_{\iota}\frac{N_{l}N^{jl}}{\lambda_{k}-\lambda_{l}}\nu}}=1;3\neq kk-NkN^{j}kV_{k})$where$V_{k}= \lambda^{8}\lambda_{k}k^{+\sum F_{j}}3j=03\lambda_{k}^{\mathrm{j}}+\frac{3}{4}\sum_{kl=1,\neq)}^{3}.\frac{1}{(\lambda_{k}-\lambda_{l})^{2}}$Now we define$\overline{I\mathrm{f}}_{\mathrm{j}}(\mathrm{j}=1,2,3)$.

by$\overline{K}_{\mathrm{j}}=K_{j}-\frac{1}{2}(s_{3}3+s1s3+\frac{1}{2}\mathit{8}22)\delta_{j1^{-\frac{3}{8}}}s2\mathit{8}^{2}3\delta_{j2^{-}}\frac{1}{4}s_{2}\delta_{j3}$where$\delta_{i\mathrm{j}}$$(j=1,2,3)$ ,is the Kronecker’s delta.

Proposition 3.3.

The transformation$(_{S,\lambda,\mu}, H)arrow(s, \lambda, \nu,\overline{K})$is canonical. where$\nu=(\nu_{1}, \nu_{2}, \nu_{3}),$$(j, k=1,2,3)$$\lambda=(\lambda_{1}, \lambda_{2,3}\lambda)_{f}\mu=(\mu_{1}, \mu_{2},\mu_{3}),$$\overline{K}=(\overline{I\mathrm{f}}_{1},\overline{IC}2,\overline{I\zeta}3)f$and $s=(s_{1}, s_{23}, S)$ .

$H=(H_{1}, H_{2}, H_{3})$,27

The $A_{3}(x)$ system.

4

In this section, we will prove Main Theorem. By means of Propositions 1.2,
1.3 and 3.3, it suffices to establish the following theorem:
The conditions (1.3), (1.4) of the complete integrability
Theorem 4.1.
are equivalent to the following completely integrable
s.ystem ofpartial differential equations:$Hamilt_{\mathit{0}}nia,n$$(\overline{K})$$\frac{\partial\lambda_{k}}{\partial s_{j}}=\frac{\partial\overline{K}_{j}}{\partial\nu_{k-}}$$(k,j=1,2,3)$ .

$\frac{\partial\nu_{k}}{\partial s_{j}}=-\frac{\partial\overline{K}_{j}}{\partial\lambda_{k}}$Lemma 4.1.

whereables,$\sigma_{j}$$\lambda_{1},$,$A_{1}(x)$$=$$\frac{1}{2}\frac{1}{\Lambda(x)}$$A_{3}(x)$$=$$\frac{1}{2}(x^{2}-\sigma 1X+\sigma_{2})\frac{1}{\Lambda(x)}$$A_{2}(x)= \frac{1}{2}(x-\sigma_{1})\frac{1}{\Lambda(x)}$.

denotes the j-th elementary symmetric polynomials of three vari$\lambda_{2},$$\lambda_{3}$Lemma 4.2.

.

) is complete integrable.

$\overline{(K}$The polynomial Hamiltonian structure.

5

Theorem 5.1.
the variables
(5.1)

(5.2)

For the Hamiltonian$sy_{\mathit{8}t}em(H)$, define the changes of$(j=1,2,3)$ .

$=$$\overline{\sigma}_{j}=(-1)^{\mathrm{j}-1}\sigma_{j}$,$H_{j}=R_{j}$The $change\mathit{8}(5.1)and(5.2)$ take $(H)$ to the system:
$(R)$

,

$\frac{\theta\overline{\sigma}_{k}}{\partial s_{j}}=\frac{\partial R_{j}}{\partial\rho_{k}}$$\frac{\partial\rho_{k}}{\partial s_{j}}=-\frac{\partial R_{j}}{T\sigma_{k}}$.

28with the polynomial Hamiltonian of the $fom$ :(5.3)$\vec{R}=\Sigma^{(}arrow 2)(\overline{\rho}+\Sigma 1\rho 2)arrow \mathrm{t})\prec 1)+\Sigma^{\mathrm{t}^{0}}\sim)\rho^{\wedge 0)}+(\alpha+\frac{1}{2})\Sigmaarrow$

.

where
$\rho^{\triangleleft 2)}=$,$\rho^{\triangleleft 1)}=$,$\rho^{\triangleleft 0)}=$,$\tilde{R}=$$\Sigma^{(2)}=\frac{1}{2}arrow$ ,$\Sigma^{\langle 1)}=arrow$ ,$\Sigma^{\langle 0)}=-arrow\frac{1}{2}$

,

where

”

.

,

$\alpha_{11}$$=$$2\overline{\sigma}_{1^{2}}+2\overline{\sigma}_{2}+3\mathit{8}3$$\alpha_{12}$$=$$.2\overline{\sigma}_{1}\overline{\sigma}_{2}+2\overline{\sigma}_{3}+2s_{2},$$\alpha_{13}$$=$$2 \overline{\sigma}1\overline{\sigma}3+S1+\frac{3}{4}S^{2}3$$\alpha_{21}$$=$$2\overline{\sigma}_{3}+2\overline{\sigma}1\overline{\sigma}2+2s_{2}$$\alpha_{22}$$=$$2 \overline{\sigma}_{2^{2}}+3_{S}3\overline{\sigma}2-2_{S_{2}\overline{\sigma}}1+\mathit{8}_{1}+\frac{3}{4}S^{2}3$$\alpha_{23}$$=$$2 \overline{\sigma}_{2}\overline{\sigma}_{3}+3S3\overline{\sigma}3^{-}(S_{1}+\frac{3}{4}S_{3}2)\overline{\sigma}_{1}+1$$\alpha_{31}$$=$$2 \overline{\sigma}_{1}\overline{\sigma}_{3}+S_{1}+\frac{3}{4}S^{2}3$$\alpha_{32}$$=$$2\overline{\sigma}_{2}\overline{\sigma}_{3}+ss_{3}\overline{\sigma}_{3}-(_{S_{1}+\frac{3}{4}}s32)\overline{\sigma}_{1}+2$$\alpha_{33}$$=$$2\overline{\sigma}_{3^{2}}+2s_{23^{-}}\overline{\sigma}(_{S_{1}+\frac{3}{4}S^{2})}3\overline{\sigma}_{2}-\overline{\sigma}_{1}$$P$$\prime 1\backslash \cdot$$-$$$\mathrm{a}.,$

.

’

,
’

’

,

.

,
,

29

$\check{\Sigma}=-$Next we define$E_{1},$ $E_{2},$ $E_{3}$.

by$–+( \alpha+\frac{1}{2})$ .
The Hamiltonian system $(R)$ is equivalent to the system:

,

$(E)$

$(j, k=1,2,3)$ .

$\frac{\partial\rho_{k}}{\partial_{\backslash ^{8}j}}=-\frac{\partial E_{j}}{\partial\overline{\sigma}_{k}}$$\frac{\partial\overline{\sigma}_{k}}{\partial s_{j}}=\frac{\partial E_{j}}{\partial\rho_{k}}$Furthermore we define the change of the variables as follows:$\overline{\sigma}_{1}=q_{1}$

(6.4)

,

$\overline{\sigma}_{2}=q_{2}-t3$,$\overline{\sigma}_{3}=q_{3}+\frac{1}{2}t_{3}q_{1}-\frac{1}{2}t_{2}$$\rho_{1}=p_{1^{-}}\frac{1}{2}t_{3}p_{3}$(6.5)$\rho_{2}=p_{2}$$\rho_{3}=p_{3}$$s_{2}=t_{2}$$s_{3}=t_{3}$$\overline{L}_{1}=L_{\mathrm{i}}$

(6.7)

,

,
,

$s_{1}=t_{1^{-\frac{1}{4}}}t_{3}^{2}$

(6.6)

,

,

,
,

,

$\overline{L}_{2}=L_{2}-\frac{1}{\int}p_{3}$,$\overline{L}_{3}=L_{3}+\overline{2}^{t}3L1-p,2+\frac{1}{2}q1p_{3}$.

30

By simple computation, we see that the change of the variables defined by(6.4), (6.5), (6.6), (6.7) is a canonical transformation of the Hamiltoniansystem $(E)$ . which takes the Hamiltonian system $(E)$ to the Hamiltoniansystem $(L)$ :$(L)$$\frac{\partial q_{k}}{\partial t_{j}}=\frac{\partial L_{j}}{\partial p_{k}}$,$(j, k=1,2,3)$ .

$\frac{\partial p_{k}}{\partial t_{j}}=-\frac{\partial L_{j}}{\partial q_{k}}$with the polynomial Hamiltonian:(6.9)$\vec{L}=\vec{\mathrm{Y}}^{(2)}\vec{P}(2)+\vec{Y}^{(1)}\vec{P}\langle 1)+\vec{Y}^{(0)}\vec{P}\mathrm{t}0\rangle+(\alpha+\frac{1}{2})\vec{\mathrm{Y}}$

.

where
$\vec{L}=$ ,$0$$\vec{\mathrm{Y}}^{\langle 2\rangle}=(_{\frac{1}{2}}^{0}$$\vec{P}^{(2)}=(_{p_{3}^{2}}^{p^{2}}p_{2}^{2}1)$$\frac{1}{2}q_{1}^{2^{-q_{1}}}-\frac{1}{4}t\frac{1}{2}3$,$\vec{P}^{(1)}=$ ,$\vec{P}^{(0)}=$$\frac{1}{2}q_{2^{-}}^{2}\frac{1}{2}q_{1}q\frac{1}{2}q_{1}q_{2}\frac{1}{4}t3q_{1}+\frac{1}{2}q_{1^{-}}3-\frac t_{2}3^{-}\frac{+1}{4}t3q_{1}+2\frac{1}{4}t_{2qt_{3}}\frac{411}{4}q2+-\frac{1}{2}q_{2}\frac{1}{8}t^{2}3$$\vec{\mathrm{Y}}^{\langle 1)}=(_{-q_{1}}^{0}1$$q_{1}q_{2}+ \frac{1}{2}t_{3^{-q1}}q_{1}q3-q^{2}1-\frac{1}{+2}t_{3}\frac{1}{2}t_{2}$$-q_{2}-q_{1}1$$(i,j=1,2,3)$ ,$\vec{\mathrm{Y}}^{\{0)}=-((\mathrm{Y}_{ij})\mathrm{I}$,$\mathrm{Y}_{11}$$=$$q_{1}^{2}+q_{2}+ \frac{1}{2}t3$$Y_{12}$$=$$q_{1}q_{2}- \frac{1}{2}t_{3}q_{1}+q3+\frac{1}{2}t_{2}$$\mathrm{Y}_{13}$$=$$q_{1}q_{3}- \frac{1}{2}t_{2q_{1^{-\frac{1}{2}}}}t3q_{2}+\frac{1}{2}t_{1}$$Y_{21}$$=$$q_{1}q_{2^{-\frac{1}{2}t_{3}q}}q_{1}+3+ \frac{1}{2}t_{2}$

,

,

,

),

,

),

31

$Y_{22}$$=$$q_{2}^{2}-t2q_{1}- \frac{1}{2}t_{3}q2+\frac{1}{2}t1-\frac{1}{4}t^{2}3$$\mathrm{Y}_{23}$$=$$q_{2}q_{3}+( \frac{1}{4}t_{3}^{2}-\frac{1}{2}t1)q1^{-}\frac{1}{2}t_{2}q2-\frac{1}{2}t_{2}t3$$Y_{31}$$=$$q_{1}q_{3}- \frac{1}{2}t2q_{1}-\frac{1}{2}t3q_{2}+\frac{1}{2}t_{1}$$Y_{32}$$=$$q_{2}q_{3}+( \frac{1}{4}t_{3}^{2}-\frac{1}{2}t_{1})q_{1^{-}}\frac{1}{2}t2q2-\frac{1}{2}t2t_{3}$

Y33

$=$

$q_{3}^{2}+ \frac{1}{2}t_{2}t3q_{1}-\frac{1}{2}t_{1}q_{2}-\frac{1}{4}t^{2}2+\frac{1}{8}t_{3}^{3}$

’

,

,
,

.

$\vec{Y}=-$ .

Particular solution of system $(L)$ .

6

in $(L)$ .
Theorem 6.1. Suppose that
1) The Hamiltonian system $(L)$ admits a solution of the form:$\alpha=-\frac{1}{2}$(6.1)$(q_{1}, q_{2}, q_{3},p1,p_{2},p_{3})=(q_{1}(t), q2(t),$

(6.2)

2)

$q_{j}(t)= \frac{\partial}{\partial t_{j}}\log u$$u(t)$$q3(t),$$\mathrm{o},$$0,0)$.

$(j=1,2,3)$ .

is a general solution of the following system of partial differential

32

equations:$\frac{\partial^{2}u}{\partial t_{1}^{2}}=-\frac{\partial u}{\partial t_{2}}-\frac{1}{2}t_{3}u$,$\frac{\partial^{2}u}{\partial t_{1}\partial t_{2}}=\frac{1}{2}t_{3^{\frac{\partial u}{\partial t_{1}}}}-\frac{\partial u}{\partial t_{3}}-\frac{1}{2}t_{2}u$,$\frac{\partial^{2}u}{\partial t_{1}\partial t_{3}}=\frac{1}{2}t_{2}\frac{\partial u}{\partial t_{1}}+\frac{1}{2}t\mathrm{s}\frac{\partial u}{\partial t_{2}}-\frac{1}{2}t_{1}u$

,

(6.3)

,

$\frac{\partial^{2}u}{\partial t_{2}^{2}}=t_{2}\frac{\partial u}{\partial t_{1}}+\frac{1}{2}t_{3^{\frac{\partial u}{\partial t_{2}}}}-(\frac{1}{2}t1-\frac{1}{4}t_{3}^{2})u$$\frac{\partial^{2}u}{\partial t_{2}\partial t_{3}}=(\frac{1}{2}t_{1}-\frac{1}{4}t_{3}^{2})\frac{\partial u}{\partial t_{1}}+\frac{1}{2}t2^{\frac{\partial u}{\partial t_{2}}}+\frac{1}{2}t2t_{3}u$$\frac{\partial^{2}u}{\partial t_{3}^{2}}=-\frac{1}{2}t_{2}t_{3}\frac{\partial u}{\partial t_{1}}+\frac{1}{2}t_{1^{\frac{\partial u}{\partial t_{2}}}}+\frac{1}{4}(t-2\frac{1}{2}2t)3u3$

Lemma 6.2. If
solution of the form:$\alpha=-\frac{1}{2}$$q_{j}(t)$.

, the Hamiltonian $\mathit{8}ystem(L)$ admits a particular$(q_{1}, q_{2}, q_{3},p1,p2,p_{3})=(q_{1}(t), q_{2}(t),$ $q3(t),$

with

,

$\mathrm{o},$$0,0)$.

$(j=1,2,3)$ determined by following equations.
$\frac{\partial q_{k}}{\partial t_{j}}=-Y_{jk}(t, q)$$(k,j=1,2,3)$ .

Proposition 6.3. For the system.
(1) The dimension of solution space over is 4.
(2) The base of the solution of (6.3) is given by$(6.3^{\iota})$$\mathrm{C}$$u_{i}(t)= \int_{r:}\exp[-\frac{2}{5}\lambda^{5}-t3\lambda^{3}-t2\lambda^{2}-(t_{1}+\frac{1}{2}t^{2})3\lambda-\frac{1}{2}t2t_{3}]d\lambda$

where$r_{i}$

are paths in the complex plane desc

$7^{\cdot}ibed$$(i=1,2,3,4)$ .

in Figure 1.

References[1] Okamoto, K., Isomonodromic deformation and Painlev\’e equations andthe Garnier system, J. Fav. Aci. Univ. Tokyo Sec. IA, Math. 33(1986),

575-618.

33[2] Kimura, H., The degeneration of the two dimensional Gamier systemand the polynomial Hamiltonian structure, Ann. Mat. Pura. Appl., CLV(1989), P. 25-34. :.

Iin$\lambda$$\mathrm{r}_{1}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}$1: