$d\hat{Y}=\left[\sum_{k=1}^{n}\hat{B}^{(k)}dx_{k}\right]\hat{Y}.$ (111)

###### Proof.

Put $P_{k}(x)=(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}$ for
$k=1,2,\dots n$. Then we have

$\displaystyle(dM_{\tilde{V}})M_{\tilde{V}}^{-1}$
$\displaystyle=\begin{pmatrix}du_{1}&du_{2}&\dots&du_{n}\end{pmatrix}\begin{pmatrix}{\partial
M_{\tilde{V}}\over\partial u_{1}}&{\partial M_{\tilde{V}}\over\partial
u_{2}}&\dots&{\partial M_{\tilde{V}}\over\partial
u_{n}}\end{pmatrix}^{t}M_{\tilde{V}}^{-1}$
$\displaystyle=\begin{pmatrix}du_{1}&du_{2}&\dots&du_{n}\end{pmatrix}M_{\tilde{V}}^{-1}\begin{pmatrix}\tilde{V}_{1}\,M_{\tilde{V}}&\tilde{V}_{2}\,M_{\tilde{V}}&\dots&\tilde{V}_{n}\,M_{\tilde{V}}\end{pmatrix}^{t}M_{\tilde{V}}^{-1}$
$\displaystyle=\begin{pmatrix}dx_{1}&dx_{2}&\dots&dx_{n}\end{pmatrix}\left({\partial
u\over\partial
x}\right)M_{\tilde{V}}^{-1}\begin{pmatrix}P_{1}(x)&P_{2}(x)&\dots&P_{n}(x)\end{pmatrix}^{t}$
$\displaystyle=\begin{pmatrix}dx_{1}&dx_{2}&\dots&dx_{n}\end{pmatrix}M_{V}^{-1}\begin{pmatrix}P_{1}(x)&P_{2}(x)&\dots&P_{n}(x)\end{pmatrix}^{t}.$

Consequently $\hat{B}^{(k)}=\sum_{j=1}^{n}(M_{V}^{-1})_{k,j}\,P_{j}(x)$. Since
$\det(M_{V})=c\,h(x)$, $c\in\mathbb{C}\setminus\\{0\\}$, we find
$h(x)\hat{B}^{(k)}(x)\in\mathbb{C}[x]^{n\times n}$.

Since $y(u)$ is a solution of

$d\begin{pmatrix}\partial_{u_{1}}&\partial_{u_{2}}&\dots&\partial_{u_{n}}\end{pmatrix}^{t}y=0,$
(112)

and since

$\displaystyle\begin{pmatrix}\partial_{u_{1}}&\partial_{u_{2}}&\dots&\partial_{u_{n}}\end{pmatrix}^{t}=$
$\displaystyle\left({\partial x\over\partial
u}\right)\begin{pmatrix}\partial_{x_{1}}&\partial_{x_{2}}&\dots&\partial_{x_{n}}\end{pmatrix}^{t}$
$\displaystyle=$ $\displaystyle\left({\partial x\over\partial
u}\right)M_{V}^{-1}\begin{pmatrix}V_{n}&V_{n-1}&\dots&V_{1}\end{pmatrix}^{t},$

$y(u)$ satisfies

$\left[d-(dM_{\tilde{V}})M_{\tilde{V}}^{-1}\right]\begin{pmatrix}V_{n}&V_{n-1}&\dots&V_{1}\end{pmatrix}^{t}y=0.$

This proves the lemma. ∎

######  Lemma 6.5.

Let $\hat{Y}$, $\hat{B}^{(k)}$ be the same as in Lemma 6.4. Then there is an
upper triangular matrix $P(x^{\prime})\in\mathbb{C}[x]^{n\times n}$ such that,
if we put

$Y_{0}=(y_{1},y_{2},\dots,y_{n})^{t}=P(x^{\prime})\hat{Y},\quad
B^{(k)}_{0}=P\hat{B}^{(k)}P^{-1},\ 1\leq k\leq n,$

then the differential equation

$dY_{0}=\left[\sum_{k=1}^{n}B^{(k)}_{0}\,dx_{k}\right]Y_{0}$ (113)

satisfied by $Y_{0}$ is an Okubo type system with $(B_{0}^{(k)})_{\epsilon}=0$
(See Lemma 2.1 for the $\mathcal{E}$-part $(B_{0}^{(k)})_{\epsilon}$ of
$B_{0}^{(k)}$), and the residue matrix $(B_{0})_{\infty}$ of
$B_{0}^{(n)}dx_{n}$ at $x_{n}=\infty$ is the same as (100). The matrix
$P(x^{\prime})$ can be chosen such that all the components are weighted
homogeneous with $w(P(x^{\prime})_{i,j})=w(\hat{y_{i}})-w(\hat{y_{j}})$, and
that $P(0)=I_{n}$. In particular, it holds that
$y_{n}=\hat{y}_{n}=(1/d_{n})y(u)$.

###### Proof.

Let $\hat{B}_{\infty}(x^{\prime})=-(x_{n}\hat{B}^{(n)})|_{x_{n}=\infty}$ be
the residue matrix of $\hat{B}^{(n)}dx_{n}$ at $x_{n}=\infty$. From

$w(\hat{B}^{(k)}_{i,j}(x^{\prime}))=w(\hat{y}_{i})-w(\hat{y}_{j})-w(x_{k})=w(x_{j})-w(x_{i})-w(x_{k}),$
(114)

we find that the degree of $h(x)\hat{B}^{(n)}_{i,j}(x)$ in $x_{n}$ is at most
$n-1$, which implies that $-\hat{B}_{\infty}$ is the coefficients of
$x_{n}^{n-1}$ of $h(x)\hat{B}^{(n)}(x)$. Consequently
$(\hat{B}_{\infty})_{i,j}(x^{\prime})$ is a weighted homogeneous polynomial in
$x^{\prime}$ with

$w((\hat{B}_{\infty})_{i,j}(x^{\prime}))=w(x_{j})-w(x_{i}),$

and hence $\hat{B}_{\infty}(x^{\prime})$ is an upper triangular matrix.

Since

$\left(\sum_{k=1}^{n}w(x_{k})x_{k}\hat{B}^{(k)}(x)\right)\hat{Y}=\left(\sum_{k=1}^{n}w(x_{k})x_{k}{\partial\over\partial
x_{k}}\right)\hat{Y}=\mbox{diag}[w(\hat{y}_{1}),w(\hat{y}_{2}),\dots,w(\hat{y}_{n})]\hat{Y},$

for all solutions $\hat{Y}$ of (111), we have

$\sum_{k=1}^{n}w(x_{k})x_{k}\hat{B}^{(k)}(x)=\mbox{diag}[w(\hat{y}_{1}),w(\hat{y}_{2}),\dots,w(\hat{y}_{n})].$

In particular, we have

$(x_{n}\hat{B}^{(n)}(x))|_{x^{\prime}=0}=\mbox{diag}[w(\hat{y}_{1}),w(\hat{y}_{2}),\dots,w(\hat{y}_{n})],$
(115)

which shows $(\hat{B}_{\infty})_{i,i}=-w(\hat{y}_{i})$.

Since $w(\hat{y}_{i})\neq w(\hat{y}_{j})$ for $i\neq j$, there is an upper
triangular matrix $P(x^{\prime})\in\mathbb{C}[x^{\prime}]^{n\times n}$ such
that

$\displaystyle P(0)=I_{n},\quad
w(P(x^{\prime})_{i,j})=w((\hat{B}_{\infty})_{i,j}),$ $\displaystyle
P(x^{\prime})\hat{B}_{\infty}(x^{\prime})P(x^{\prime})^{-1}=-\mbox{diag}[w(\hat{y}_{1}),w(\hat{y}_{2}),\dots,w(\hat{y}_{n})].$

From $B^{(n)}_{0}(x)=P(x^{\prime})\hat{B}^{(n)}(x)P(x^{\prime})^{-1}$, we find
that

$(B_{0})_{\infty}=-(x_{n}B_{0}^{(n)})|_{x_{n}=\infty}=P(x^{\prime})\left[-(x_{n}\hat{B}^{(n)})|_{x_{n}=\infty}\right]P(x^{\prime})^{-1}=P(x^{\prime})\hat{B}_{\infty}P(x^{\prime})^{-1},$

which is equal to (100).

Since the residue matrix of $\hat{B}^{(n)}dx_{n}$ at zeros of
$h(x^{\prime},x_{n})$ is of rank one and diagonalizable, so is the residue
matrix of $B^{(n)}_{0}dx_{n}$. Consequently, by Lemma 2.4, we find that there
exists a $n\times n$ matrix $T_{0}(x^{\prime})$ such that

$B^{(n)}_{0}=-(x_{n}I_{n}-T_{0}(x^{\prime}))^{-1}B_{\infty}$ (116)

at least at any generic point $x^{\prime}$. Let

$h(x)=x_{n}^{n}-s_{1}(x^{\prime})x_{n}^{n-1}+\dots+(-1)^{n}s_{n}(x^{\prime}),\quad
h(x)B^{(n)}_{0}=\sum_{i=0}^{n-1}C_{i}(x^{\prime})x_{n}^{i},\
C_{n-1}(x^{\prime})=-B_{\infty}.$

Then, by induction, we find

$C_{n-i}(x^{\prime})=-(T_{0}(x^{\prime})^{i-1}-s_{1}(x^{\prime})T_{0}(x^{\prime})^{i-2}+\dots+(-1)^{i-1}s_{i-1}(x^{\prime}))\,B_{\infty},\quad
1\leq i\leq n.$ (117)

In particular, it holds that
$T_{0}(x^{\prime})=s_{1}(x^{\prime})I_{n}-C_{n-2}(x^{\prime})B_{\infty}^{-1}$,
which implies that all entries of $T_{0}(x^{\prime})$ are weighted homogeneous
polynomials, and the equality (116) holds at all points $x^{\prime}$. This
concludes that the equation (113) is of Okubo type.

Recall that the $\mathcal{E}$-part $(B_{0}^{(k)})_{\epsilon}$ of $B_{0}^{(k)}$
is the coefficient of $x_{n}^{n}$ of $h(x)B_{0}^{(k)}(x)$, and is diagonal.
From (114), we have
$w\left(h(x)(B_{0}^{(k)})_{i,i}(x^{\prime})\right)=n-w(x_{k})$, which implies
that $(B_{0}^{(k)})_{\epsilon}=0$. ∎

Now we prove Theorem 6.1. From now on we assume

$M_{V}=M_{V^{(h)}},\quad V_{i}=V_{i}^{(h)},\ 1\leq i\leq n,$

where $M_{V^{(h)}}$, $V_{i}^{(h)}$ are defined by (48) with respect to $h(x)$.

###### Proof of Theorem 6.1.

Let $B_{\infty}$, $\hat{Y},P,Y_{0}=P\hat{Y}$, $B^{(i)}_{0}$,
$T_{0}(x^{\prime})$ be as before. Let
$B_{\infty}=\mbox{diag}[\lambda_{1},\lambda_{2},\dots,\lambda_{n}]$. Since the
equation (113) is of Okubo type such that

$B^{(n)}=-(x_{n}I_{n}-T_{0}(x^{\prime}))^{-1}B_{\infty},\quad\mbox{and}\
(B_{0}^{(k)})_{\epsilon}=0,\ 1\leq k\leq n-1,$

there is a matrix
$C(x)=x_{n}I_{n}+C^{\prime}(x^{\prime})\in\mathbb{C}[x]^{n\times n}$ such that

$\displaystyle x_{n}I_{n}-T_{0}(x^{\prime})=V_{1}^{(h)}C(x),\quad
B^{(i)}_{0}=-(V_{1}^{(h)}C(x))^{-1}\,{\partial C(x)\over\partial
x_{i}}B_{\infty},\ 1\leq i\leq n.$

Then $C_{i,j}(x)$ are weighted homogeneous polynomials with
$w(C_{i,j})=1-w(x_{i})+w(x_{j})$. In particular, $w(C_{n,j})=w(x_{j})$.

Let $\hat{A^{(i)}}$ and $A^{(i)}$ be the matrices such that
$V^{(h)}_{i}\hat{Y}=\hat{A}^{(i)}\hat{Y}$ and $V^{(h)}_{i}Y_{0}=A^{(i)}Y_{0}$
hold respectively. Recall $A^{(i)}=-\tilde{B}^{(n-i+1)}B_{\infty}$ (see (53)),
and hence

$\partial C/\partial x_{i}=\tilde{B}^{(i)}=-A^{(n-i+1)}(B_{\infty})^{-1}.$
(118)

Comparing the $n$-th row of $\hat{A}^{(n-i+1)}\hat{Y}=V^{(h)}_{n-i+1}\hat{Y}$,
we have

$\sum_{j=1}^{n}\hat{A}^{(n-i+1)}_{n,j}\hat{y}_{j}=V_{n-i+1}^{(h)}\hat{y}_{n}=V_{n-i+1}^{(h)}(V_{1}^{(h)}y)={1\over
d_{n}}V_{n-i+1}^{(h)}(y)={1\over d_{n}}\hat{y}_{i},$

which implies

$\hat{A}^{(n-i+1)}_{n,j}=\begin{cases}1/d_{n}&\mbox{if}\ j=i,\\\
0&\mbox{otherwise}.\end{cases}$

Since $A^{(n-i+1)}=(P\hat{A}^{(n-i+1)}+V^{(h)}_{n-i+1}P)P^{-1}$, we find

$A^{(n-i+1)}_{n,j}=\begin{cases}0&\mbox{if}\ j<i,\\\ 1/d_{n}&\mbox{if}\
j=i,\\\ \mbox{a polynomial with the weight}\ w(x_{j})-w(x_{i})&\mbox{if}\
j>i.\end{cases}$ (119)

From (118) and (119), we have

$C_{n,i}=-x_{i}/(d_{n}\lambda_{i})+c_{i}(x_{1},x_{2},\dots,x_{i-1})={1\over
1-d_{i}+d_{n}}x_{i}+c_{i}(x_{1},x_{2},\dots,x_{i-1}),$ (120)

where $c_{i}(x)$ is a weighted homogeneous polynomial in $x_{1},x_{2},\dots
x_{i-1}$ with $w(c_{i})=w(x_{i})$. If we choose

$F^{\prime}_{i}(u):={1\over
1-d_{i}+d_{n}}F_{i}(u)+c_{i}(F_{1}(u),F_{2}(u),\dots,F_{i-1}(u)),\quad 1\leq
i\leq n$

as a new fundamental system of $G$ invariant polynomials, and
$\\{t_{i}=F^{\prime}_{i}(u)\\}$ as a new coordinates on $X$, then, by
definition, it holds that $t_{j}=C_{n,j}$, $1\leq j\leq n$. Then Lemma 3.6 and
Lemma 4.4 assert that $-T=V_{1}C(t)=M_{V^{(h)}}(t)$, and Theorem 5.1 asserts
that $\\{t_{j},1\leq j\leq n\\}$ gives a flat coordinate system as the
independent variables of (113). Consequently we have

$\displaystyle\begin{pmatrix}V^{\prime}_{n}&V^{\prime}_{n-1}&\dots&V^{\prime}_{1}\end{pmatrix}^{t}$
$\displaystyle=-B_{\infty}^{-1}\,M_{V^{(h)}}(t)\begin{pmatrix}\partial_{t_{1}}&\partial_{t_{2}}&\dots&\partial_{t_{n}}\end{pmatrix}^{t}$
$\displaystyle=B_{\infty}^{-1}\,T\,\begin{pmatrix}\partial_{t_{1}}&\partial_{t_{2}}&\dots&\partial_{t_{n}}\end{pmatrix}^{t}$

Then, from the “$({\rm i})\Rightarrow({\rm ii})$” part of the proof of Theorem
5.2, we find

$Y_{0}=\begin{pmatrix}V^{\prime}_{n}&V^{\prime}_{n-1}&\dots&V^{\prime}_{1}\end{pmatrix}^{t}y_{n}.$

This completes the proof of the existence of a flat generator system of
$G$-invariant polynomials.

We next prove the uniqueness of a flat generator system of $G$-invariant
polynomials. So let $\\{F^{\prime\prime}_{i}(u)\\}$ be another flat generator
system, and assume that the coordinates $s_{i}=F^{\prime\prime}_{i}(u)$,
$1\leq i\leq n$ satisfy the properties (i) and (ii) of the theorem. Let
$V_{i}^{(h)}(t)$, $V_{i}^{(h)}(s)$, $1\leq i\leq n$ be the logarithmic vector
fields along $D$ defined by (48) in the coordinates $t$, $s$, respectively.
Let $Y_{0}^{(t)},Y_{0}^{(s)}$ be the vectors defined by (101). From (ii) of
Lemma 3.11, there is an upper triangular matrix $G(t^{\prime})\in
GL(n,\mathbb{C}[t^{\prime}])$ such that

$B_{\infty}^{-1}\begin{pmatrix}V^{(h)}_{n}(s)&V^{(h)}_{n-1}(s)&\dots&V^{(h)}_{1}(s)\end{pmatrix}^{t}=G(t^{\prime})B_{\infty}^{-1}\begin{pmatrix}V^{(h)}_{n}(t)&V^{(h)}_{n-1}(t)&\dots&V^{(h)}_{1}(t)\end{pmatrix}^{t},$

and hence $Y_{0}^{(s)}=G(t^{\prime})Y_{0}^{(t)}$. Since $Y_{0}^{(t)}$ and
$Y_{0}^{(s)}$ satisfy Okubo type equations of the form of (101),
$G(t^{\prime})$ must be diagonal matrix with constant entries. This implies
that $s_{i}=c_{i}t_{i}$, $1\leq i\leq n$ for some constant numbers $c_{i}$.

This proves the uniqueness, and completes the proof of the theorem. ∎

######  Corollary 6.1.

The potential vector field $\vec{g}=(g_{1},\dots,g_{n})$ corresponding to the
$G$-quotient system (102) is a polynomial in $t_{1},\dots,t_{n}$. Namely, the
potential vector field of the $G$-quotient system yields a polynomial solution
of the extended WDVV equation.

###### Proof.

By the above construction, $h(t)$ and $T=-M_{V^{(h)}}(t)$ are polynomials in
$t_{1},\dots,t_{n}$. Thus, $C(t)$ and $\vec{g}$ are also polynomials. ∎

######  Remark 6.2.

Irreducible finite complex reflection groups are classified by Shephard-Todd
[36]: There is an infinite family $G(de,d,r)$, plus 34 exceptional groups
$G_{4},G_{5},\dots,G_{37}$. Concrete forms of potential vector fields for the
exceptional groups are discussed in [20].

##  7 Examples of potential vector fields related to algebraic solutions of
the Painlevé VI equation

In this section we show some examples of potential vector fields in three
variable case which are related to algebraic solutions of the Painlevé VI
equation.

Algebraic solutions of the Painlevé VI equation were studied and constructed
by many authors including N. J. Hitchin [15], [16], B. Dubrovin [10], B.
Dubrovin -M. Mazzocco [11], P. Boalch [2], [3], [4], [5], A. V. Kitaev [23],
[24], A. V. Kitaev -R. Vidunas [25], [39], K. Iwasaki [17]. The classification
of algebraic solutions of the Painlevé VI equation was achieved by Lisovvy and
Tykhyy [27]. We remark that all the algebraic solutions in the list of [27]
had previously appeared in the literature (see [6] and references therein).
One of the principal aims of our study is the determination of a flat
coordinate system and a potential vector field for each of such algebraic
solutions. In spite that this aim is still not succeeded because of complexity
of computation, we show some examples of potential vector fields. Some of the
results below are already given in [20].

To avoid the confusion, we prepare the convention which will be used in the
following. We treat the case $n=3$. Let $t=(t_{1},t_{2},t_{3})$ be a flat
coordinate system and $\vec{g}=(g_{1},g_{2},g_{3})$ denotes a potential vector
field. Let $w(t_{i})$ be the weight of $t_{i}$ and assume
$0<w(t_{1})<w(t_{2})<w(t_{3})=1$. The matrix $C$ is defined by

$C=\left(\begin{array}[]{ccc}\partial_{t_{1}}g_{1}&\partial_{t_{1}}g_{2}&\partial_{t_{1}}g_{3}\\\
\partial_{t_{2}}g_{1}&\partial_{t_{2}}g_{2}&\partial_{t_{2}}g_{3}\\\
\partial_{t_{3}}g_{1}&\partial_{t_{3}}g_{2}&\partial_{t_{3}}g_{3}\\\
\end{array}\right)$

and $T=-\sum_{j=1}^{3}w(t_{j})t_{j}\partial_{t_{j}}C$. In this section, an
algebraic solution LTn means “Solution n” in Lisovvy-Tykhyy [27], pp.156-162.

###  7.1 Algebraic solutions related with icosahedron

We treat the three algebraic solutions of Painlevé VI obtained by Dubrovin
[10] and Dubrovin-Mazzocco [11].

Icosahedral solution $(H_{3})$

In this case,

$w(t_{1})=\frac{1}{5},\>w(t_{2})=\frac{3}{5},\>w(t_{3})=1$

and there is a potential defined by

$F=\frac{t_{2}^{2}t_{3}+t_{1}t_{3}^{2}}{2}+\frac{t_{1}^{11}}{3960}+\frac{t_{1}^{5}t_{2}^{2}}{20}+\frac{t_{1}^{2}t_{2}^{3}}{6}.$
(121)

Then it follows from the definition that $g_{j}=\partial_{t_{j}}F\>(j=1,2,3)$
give the potential vector field $\vec{g}=(g_{1},g_{2},g_{3})$.

We don’t enter the details on this case. See [10], [11].

Great icosahedral solution $(H_{3})^{\prime}$

Let $(t_{1},t_{2},t_{3})$ be a flat coordinate system and their weights are
given by

$w(t_{1})=\frac{3}{5},\>w(t_{2})=\frac{4}{5},\>w(t_{3})=1.$

We introduce an algebraic function $z$ of $t_{1},t_{2}$ defined by the
relation

$t_{2}+t_{1}z+z^{4}=0.$

It is clear from the definition that $\>w(z)=\frac{1}{5}$. In this case, we
consider the algebraic function of $(t_{1},t_{2},t_{3})$ defined by

$F=\frac{t_{2}^{2}t_{3}+t_{1}t_{3}^{2}}{2}-\frac{t_{1}^{4}z}{18}-\frac{7t_{1}^{3}z^{4}}{72}-\frac{17t_{1}^{2}z^{7}}{105}-\frac{2t_{1}z^{10}}{9}-\frac{64z^{13}}{585}.$

Then we see that $F$ is a solution of the WDVV equation. Indeed, we first
define

$C=\left(\begin{array}[]{ccc}\partial_{t_{1}}\partial_{t_{3}}F&\partial_{t_{1}}\partial_{t_{2}}F&\partial_{t_{1}}^{2}F\\\
\partial_{t_{2}}\partial_{t_{3}}F&\partial_{t_{2}}^{2}F&\partial_{t_{2}}\partial_{t_{3}}F\\\
\partial_{t_{3}}^{2}F&\partial_{t_{3}}\partial_{t_{2}}F&\partial_{t_{3}}\partial_{t_{1}}F\\\
\end{array}\right).$

Then $\partial_{t_{i}}C\>(i=1,2,3)$ commute to each other. This condition is
equivalent to that $F$ is a solution of the WDVV equation.

Great dodecahedron solution $(H_{3})^{\prime\prime}$

Let $(t_{1},t_{2},t_{3})$ be a flat coordinate system and their weights are
given by

$w(t_{1})=\frac{1}{3},\>w(t_{2})=\frac{2}{3},\>w(t_{3})=1.$

We introduce an algebraic function $z$ of $t_{1},t_{2}$ defined by the
relation

$-t_{1}^{2}+t_{2}+z^{2}=0.$

It is clear from the definition that $\>w(z)=\frac{1}{3}$. In this case, we
consider the algebraic function of $(t_{1},t_{2},t_{3})$ defined by

$F=\frac{t_{2}^{2}t_{3}+t_{1}t_{3}^{2}}{2}+\frac{4063t_{1}^{7}}{1701}+\frac{19t_{1}^{5}z^{2}}{135}-\frac{73t_{1}^{3}z^{4}}{27}+\frac{11t_{1}z^{6}}{9}-\frac{16z^{7}}{35}.$

Then $F$ is also a solution of the WDVV equation.

######  Remark 7.1.

The potential $F$ for the icosahedral solution $(H_{3})$ was firstly obtained
by B. Dubrovin [10]. The above algebraic solutions including the remaining two
cases $(H_{3})^{\prime},(H_{3})^{\prime\prime}$ were treated by B. Dubrovin
and M. Mazzocco [11]. The icosahedral solution $(H_{3})$ is constructed by the
polynomial $F$ (cf. (121)). In the remaining two cases, the potentials are not
polynomials but algebraic functions. The authors don’t know whether the
potentials for the cases $(H_{3})^{\prime}$ and $(H_{3})^{\prime\prime}$ are
known or not.

###  7.2 Algebraic solution related with the complex reflection group ST24
([2])

The following case is related with the complex reflection group ST24.

Klein solution of P. Boalch [2] (LT8)

In this case,

$w(t_{1})=\frac{2}{7},\>w(t_{2})=\frac{3}{7},\>w(t_{3})=1$
$\begin{array}[]{lll}g_{1}&=&(-2t_{1}^{3}t_{2}+t_{2}^{3}+12t_{1}t_{3})/12,\\\
g_{2}&=&(2t_{1}^{5}+5t_{1}^{2}t_{2}^{2}+10t_{2}t_{3})/10,\\\
g_{3}&=&(-8t_{1}^{7}+21t_{1}^{4}t_{2}^{2}+7t_{1}t_{2}^{4}+28t_{3}^{2})/56.\\\
\end{array}$

The determinant $\det(-T)$ is regarded as the discriminant of the complex
reflection group ST24 if $t_{1},t_{2},t_{3}$ are taken as basic invariants
(cf. [36]).

###  7.3 Algebraic solutions related with the complex reflection group ST27
(cf. [36])

The following two cases are related with the complex reflection group ST27.

Solution 38 of P. Boalch [3] (LT26)

In this case,

$w(t_{1})=\frac{1}{5},\>w(t_{2})=\frac{2}{5},\>w(t_{3})=1$
$\begin{array}[]{lll}g_{1}&=&(-t_{1}^{6}-15t_{1}^{4}t_{2}+15t_{1}^{2}t_{2}^{2}+10t_{2}^{3}+30t_{1}t_{3})/30,\\\
g_{2}&=&(5t_{1}^{7}+3t_{1}^{5}t_{2}+15t_{1}^{3}t_{2}^{2}-5t_{1}t_{2}^{3}+6t_{2}t_{3})/6,\\\
g_{3}&=&(-105t_{1}^{10}+200t_{1}^{8}t_{2}+350t_{1}^{6}t_{2}^{2}+175t_{1}^{2}t_{2}^{4}-14t_{2}^{5}+20t_{3}^{2})/40.\\\
\end{array}$

The determinant $\det(-T)$ is regarded as the discriminant of the complex
reflection group ST27, if $t_{1},t_{2},t_{3}$ are taken as basic invariants.

Solution 37 of P. Boalch [3] (LT27)

In this case, $z$ is an algebraic function of $t_{1},t_{2}$ defined by

$-t_{2}-t_{1}z+2z^{3}=0.$
$w(t_{1})=\frac{2}{5},\>w(t_{2})=\frac{3}{5},\>w(t_{3})=1,\>w(z)=\frac{1}{5}$
$\begin{array}[]{lll}g_{1}&=&(175t_{1}t_{3}-70t_{1}^{3}z+70t_{1}^{2}z^{3}+378t_{1}z^{5}-540z^{7})/175,\\\
g_{2}&=&(10t_{1}^{4}-120t_{1}t_{2}^{2}+75t_{2}t_{3}+30t_{1}^{2}z^{4}-192t_{1}z^{6}+324z^{8})/75,\\\
g_{3}&=&(16t_{1}^{5}+80t_{1}^{2}t_{2}^{2}+25t_{3}^{2}-80t_{1}^{3}z^{4}+540t_{1}^{2}z^{6}-1080t_{1}z^{8}+432z^{10})/50.\\\
\end{array}$

The determinant $\det(-T)$ is regarded as the discriminant of the complex
reflection group ST27, if $z,t_{1},t_{3}$ are taken as basic invariants.

###  7.4 Algebraic solutions related with the polynomial $F_{B_{6}}$ in [34]

We recall the polynomial

$F_{B_{6}}=9xy^{4}+6x^{2}y^{2}z-4y^{3}z+x^{3}z^{2}-12xyz^{2}+4z^{3}$

which is a defining equation of a free divisor in $\mathbb{C}^{3}$ (cf. [34]).
There are two algebraic solutions which are related with the polynomial
$F_{B_{6}}$.

Solution 27 of Boalch [3] (LT13)

In this case,

$w(t_{1})=\frac{1}{15},\>w(t_{2})=\frac{1}{3},\>w(t_{3})=1$
$\begin{array}[]{lll}g_{1}&=&-\frac{1}{33}t_{1}(3t_{1}^{10}t_{2}+11t_{2}^{3}-33t_{3}),\\\
g_{2}&=&\frac{1}{76}(-5t_{1}^{20}+114t_{1}^{10}t_{2}^{2}+19t_{2}^{4}+76t_{2}t_{3}),\\\
g_{3}&=&\frac{1}{870}(100t_{1}^{30}+1740t_{1}^{20}t_{2}^{2}-5220t_{1}^{10}t_{2}^{4}+116t_{2}^{6}+435t_{3}^{2}).\\\
\end{array}$

The determinant $\det(-T)$ coincides with $F_{B_{6}}$ by a weight preserving
coordinate change up to a non-zero constant factor.

Solution obtained by A. Kitaev [24] (LT14)

In this case, $z$ is an algebraic function of $t_{1},t_{2}$ defined by

$t_{1}^{2}+t_{2}z^{6}+z^{16}=0.$
$w(t_{1})=\frac{8}{15},\>w(t_{2})=\frac{2}{3},\>w(t_{3})=1,\>w(z)=\frac{1}{15}$
$\begin{array}[]{lll}g_{1}&=&-(2093t_{1}^{4}-897t_{1}t_{3}z^{9}+3450t_{1}^{2}z^{16}+525z^{32})/(897z^{9}),\\\
g_{2}&=&(-238t_{1}^{5}+85t_{2}t_{3}z^{15}+1700t_{1}^{3}z^{16}-750t_{1}z^{32})/(85z^{15}),\\\
g_{3}&=&(49t_{1}^{6}+2415t_{1}^{4}z^{16}+3t_{3}^{2}z^{18}+795t_{1}^{2}z^{32}-35z^{48})/(6z^{18}).\\\
\end{array}$

The determinant $\det(-T)$ regarded as a polynomial of $z^{5},t_{2},t_{3}$
coincides with $F_{B_{6}}$ by a weight preserving coordinate change up to a
non-zero constant factor.

###  7.5 Algebraic solutions related with the polynomial $F_{H_{2}}$ in [34]

We recall the polynomial

$F_{H_{2}}=100x^{3}y^{4}+y^{5}+40x^{4}y^{2}z-10xy^{3}z+4x^{5}z^{2}-15x^{2}yz^{2}+z^{3}$

which is a defining equation of a free divisor in $\mathbb{C}^{3}$ (cf. [34]).
There are two algebraic solutions which are related with the polynomial
$F_{H_{2}}$.

Solution 29 of P. Boalch [3] (LT18)

In this case,

$w(t_{1})=\frac{1}{10},\>w(t_{2})=\frac{1}{5},\>w(t_{3})=1$
$\begin{array}[]{lll}g_{1}&=&-t_{1}(5t_{1}^{6}t_{2}^{2}-14t_{2}^{5}-2t_{3})/2,\\\
g_{2}&=&(5t_{1}^{12}+275t_{1}^{6}t_{2}^{3}-55t_{2}^{6}+33t_{2}t_{3})/33,\\\
g_{3}&=&(-100t_{1}^{18}t_{2}+2550t_{1}^{12}t_{2}^{4}+12750t_{1}^{6}t_{2}^{7}+595t_{2}^{10}+9t_{3}^{2})/18.\\\
\end{array}$

The determinant $\det(-T)$ regarded as a polynomial of $t_{2},t_{1}^{6},t_{3}$
coincides with $F_{H_{2}}$ by a weight preserving coordinate change up to a
non-zero constant factor.

Solution 30 of P. Boalch [3] (LT19)

In this case, $z$ is an algebraic function of $t_{1},t_{2}$ defined by

$t_{1}^{6}+t_{2}z^{6}+z^{9}=0.$
$w(t_{1})=\frac{3}{10},\>w(t_{2})=\frac{3}{5},\>w(t_{3})=1,\>w(z)=\frac{1}{5}$
$\begin{array}[]{lll}g_{1}&=&t_{1}(-80t_{2}^{2}+910t_{3}z+165t_{2}z^{3}+63z^{6})/(910z),\\\
g_{2}&=&(4t_{2}t_{3}-12t_{2}^{2}z^{2}-36t_{2}z^{5}-27z^{8})/4,\\\
g_{3}&=&(-560t_{1}^{18}+595t_{3}^{2}z^{17}+7140t_{2}^{2}z^{21}-8160t_{2}z^{24}-15113z^{27})/(1190z^{17}).\\\
\end{array}$

The determinant $\det(-T)$ regarded as a polynomial of $z,t_{2},t_{3}$
coincides with $F_{H_{2}}$ by a weight preserving coordinate change up to a
non-zero constant factor.

###  7.6 Algebraic solution related with $E_{14}$-singularity

Solution 13 of P. Boalch [4] (LT30)

In this case,

$w(t_{1})=\frac{1}{8},\>w(t_{2})=\frac{3}{8},\>w(t_{3})=1$
$\begin{array}[]{lll}g_{1}&=&(5t_{1}^{9}-84t_{1}^{6}t_{2}-210t_{1}^{3}t_{2}^{2}+140t_{2}^{3}+9t_{1}t_{3})/9,\\\
g_{2}&=&(140t_{1}^{11}-165t_{1}^{8}t_{2}+924t_{1}^{5}t_{2}^{2}+770t_{1}^{2}t_{2}^{3}+11t_{2}t_{3})/11,\\\
g_{3}&=&(-95680t_{1}^{16}-432320t_{1}^{13}t_{2}+780416t_{1}^{10}t_{2}^{2}-58240t_{1}^{7}t_{2}^{3}+1019200t_{1}^{4}t_{2}^{4}\\\
&&+203840t_{1}t_{2}^{5}+39t_{3}^{2})/78.\\\ \end{array}$

The determinant $\det(-T)$ in this case coincides with the polynomial

$\begin{array}[]{lll}F_{E_{14}}&=&-4x^{6}y^{6}-\frac{20}{3}x^{3}y^{7}-3y^{8}+30x^{7}y^{3}z+51x^{4}y^{4}z+24xy^{5}z-\frac{243}{4}x^{8}z^{2}\\\
&&-108x^{5}yz^{2}-56x^{2}y^{2}z^{2}-8z^{3}\\\ \end{array}$

by a weight preserving coordinate change. The polynomial $F_{E_{14}}$ is
regarded as a 1-parameter deformation of the defining polynomial of
$E_{14}$-singularity in the sense of Arnol’d. In fact

$F_{E_{14}}|_{x=0}=-3y^{8}-8z^{3}.$

For topics related to $F_{E_{14}}$, see [35], [20].

##  Appendix A Okubo type systems in several variables and isomonodromic
deformations

In this Appendix A, we show the equivalence between Okubo type systems in
several variables and the isomonodromic deformations of ordinary differential
equations of Okubo type. Let us recall that an Okubo type system in several
variables of rank $n$ is defined as follows:

$dY=\left(B^{(z)}dz+\sum_{i=1}^{n}B^{(i)}dx_{i}\right)Y,$ (122)

where
$B^{(z)}=-(zI_{n}-T)^{-1}B_{\infty},B^{(i)}=-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty}$,
and $T,B_{\infty},\tilde{B}^{(i)}$ are $n\times n$ matrices that satisfy the
assumptions (B1)-(B5) and the equations (27), (28), (29) in Section 2. Our
main aim is to prove the following:

######  Proposition A.1.

We consider (122) for $n=3$. Then there is a correspondence between Okubo type
systems in several variables of rank $3$ and solutions to the Painlevé VI
equation with the parameters
$(\theta_{0},\theta_{1},\theta_{t},\theta_{\infty})=(r_{1}-\lambda_{3},r_{2}-\lambda_{3},r_{3}-\lambda_{3},\lambda_{1}-\lambda_{2})$,
where $\lambda_{i}$ and $r_{i}$ are respectively defined in $(B2)$ and $(B5)$
in Section 2.

Here, we briefly review the theory of isomonodromic deformation (following to
[18], [19]). We consider an ordinary differential equation of rank $m$

$\frac{dY}{dz}=\sum_{i=1}^{n}\frac{B_{i}}{z-a_{i}}Y,$ (123)

where $a_{i}$ $(i=1,\dots,n)$ are mutually distinct complex numbers and
$B_{i}$ $(i=1,\dots,n)$ are $m\times m$ constant matrices. Put
$B_{\infty}:=-\sum_{i=1}^{n}B_{i}$. We assume that $B_{i}$ $(i=1,\dots,n)$ are
semisimple and $B_{\infty}$ is diagonal, namely there are invertible matrices
$G_{i}$ $(i=1,\dots,n)$ such that
$G_{i}B_{i}G_{i}^{-1}=\mbox{diag}[r_{1}^{(i)},\dots,r_{m}^{(i)}]$ and
$B_{\infty}=\mbox{diag}[r^{(\infty)}_{1},\dots,r^{(\infty)}_{m}]$. We also
assume $r_{j}^{(i)}-r_{k}^{(i)}\notin\mathbb{Z}\setminus\\{0\\}$ for
$i=1,\dots,n,\infty$, $j,k=1,\dots,m$. Take a fundamental system of solutions
$Y^{(\infty)}=Y^{(\infty)}(z)$ to (123) normalized around $z=\infty$ as
follows:

$Y^{(\infty)}=\hat{Y}^{(\infty)}(z^{-1})z^{-B_{\infty}},$

where $\hat{Y}^{\infty}(z^{-1})$ is a convergent power series of $z^{-1}$ such
that $\hat{Y}^{(\infty)}(0)=I_{m}$. Fixing paths from $z=\infty$ to $z=a_{i}$
$(i=1,\dots,n)$, the analytic continuations of $Y^{(\infty)}$ to $z=a_{i}$
$(i=1,\dots,n)$ are described as follows:

$Y^{(\infty)}=G_{i}\hat{Y}^{(i)}(z-a_{i})\,(z-a_{i})^{\Lambda_{i}}C_{i},\ \ \
\ i=1,\dots,n,$

where $\hat{Y}^{(i)}(z-a_{i})$ is a convergent power series of $z-a_{i}$ such
that $\hat{Y}^{(i)}(0)=I_{m}$, $\Lambda_{i}=G_{i}B_{i}G_{i}^{-1}$ and $C_{i}$
is a constant invertible matrix. Now we deform (123) moving
$a_{1},\dots,a_{n}$ as variables, namely we suppose that $B_{i}$
$(i=1,\dots,n)$ are functions of $a_{1},\dots,a_{n}$. Then $Y^{(\infty)}$ is a
function of $z,a_{1},\dots,a_{n}$ and $r^{(i)}_{j},G_{i},C_{i}$ are functions
of $a_{1},\dots,a_{n}$. A deformation of (123) with variables
$a_{1},\dots,a_{n}$ is said to be an isomonodromic deformation (or monodromy
preserving deformation) if $r^{(i)}_{j}$ $(i=1,\dots,n,\infty,j=1,\dots,m)$
and $C_{i}$ $(i=1,\dots,n)$ are constants independent of $a_{1},\dots,a_{n}$.
The following fact is well known (see [18], [19] for instance):

######  Fact 1.

A deformation of (123) is isomonodromic if $B_{i}$ $(i=1,\dots,n)$ satisfy the
following system of differential equations:

$dB_{i}=\sum_{j\neq i}[B_{j},B_{i}]d\log(a_{i}-a_{j}),\ \ \ \ \ i=1,\dots,n,$
(124)

which is called the Schlesinger system.

Returning to our situation, we consider an Okubo type system in several
variables of rank $n$ (122). Let $z_{1},\dots,z_{n}$ be the roots of
$\det(zI_{n}-T)$, and decompose $B^{(z)}$ into partial fractions

$B^{(z)}=\sum_{i=1}^{n}\frac{B^{(z)}_{i}}{z-z_{i}}.$ (125)

######  Lemma A.1.

The system of equations (27), (28), (29) is equivalent to the Schlesinger
system

$dB^{(z)}_{i}=\sum_{j\neq i}[B^{(z)}_{j},B^{(z)}_{i}]d\log(z_{i}-z_{j}),\ \ \
\ \ i=1,\dots,n.$ (126)

###### Proof.

On one hand, from Lemma 2.1, (27), (28), (29) are equivalent to the
integrability condition of (122). On the other hand, it is known that the
Schlesinger system (126) is equivalent to the integrability condition of the
Pfaffian system

$dY=\sum_{i=1}^{n}B^{(z)}_{i}d\log(z-z_{i})Y.$ (127)

Therefore, it is sufficient to show that the two Pfaffian systems (122) and
(127) are equivalent to each other. Changing the independent variables
$(x_{1},\dots,x_{n})$ of (122) to $(z_{1},\dots,z_{n})$, (122) is rewritten as

$dY=-(zI_{n}-T)^{-1}\left(dz+\sum_{i,j=1}^{n}\tilde{B}^{(j)}\frac{\partial
x_{j}}{\partial z_{i}}dz_{i}\right)B_{\infty}Y.$ (128)

Then, similarly to (90) in the proof of Proposition 5.1, we have
$\sum_{j=1}^{n}\tilde{B}^{(j)}\frac{\partial x_{j}}{\partial
z_{i}}=-PE_{i}P^{-1}$ and thus

$-(zI_{n}-T)^{-1}\sum_{j=1}^{n}\tilde{B}^{(j)}\frac{\partial x_{j}}{\partial
z_{i}}B_{\infty}=PE_{i}P^{-1}B_{\infty}/(z-z_{i}).$

It is straightforward to check that $B^{(z)}_{i}=-PE_{i}P^{-1}B_{\infty}$. ∎

As a preparation to prove Proposition A.1, we quote from Appendix C in [19] (a
concise explanation can be found in [2]) the following fact on the relation
between the Painlevé VI equation and the isomonodromic deformation of linear
differential equations of rank $2$. (Notations in Fact 2 are valid only in
this part.)

######  Fact 2.

We consider a Pfaffian system of rank $2$

$dZ=\bigl{(}A(x,t)dx+B(x,t)dt\bigr{)}Z,$ (129)

where

$A(x,t)=\frac{A_{0}}{x}+\frac{A_{1}}{x-1}+\frac{A_{t}}{x-t},\ \ \
B(x,t)=-\frac{A_{t}}{x-t},$ (130)

and $A_{0},A_{1},A_{t}$ are $2\times 2$ matrices whose entries are functions
of $t$ (independent of $x$). Put $A_{\infty}:=-A_{0}-A_{1}-A_{t}$. We assume
that $A_{0},A_{1},A_{t}$ are rank $1$ matrices and that $A_{\infty}$ is a
diagonal matrix. Put $\theta_{i}:=\mbox{{\rm tr}}A_{i}$ $(i=0,1,t)$. Then
$A_{i}$ $(i=0,1,t)$ can be written as follows:

$\displaystyle A_{0}=\begin{pmatrix}z_{0}+\theta_{0}&-uz_{0}\\\
u^{-1}(z_{0}+\theta_{0})&-z_{0}\end{pmatrix},\
A_{1}=\begin{pmatrix}z_{1}+\theta_{1}&-vz_{1}\\\
v^{-1}(z_{1}+\theta_{1})&-z_{1}\end{pmatrix},\
A_{t}=\begin{pmatrix}z_{t}+\theta_{t}&-wz_{t}\\\
w^{-1}(z_{t}+\theta_{t})&-z_{t}\end{pmatrix}.$ (131)

From the assumption that $A_{\infty}$ is diagonal, we have the following
relations:

$\displaystyle uz_{0}+vz_{1}+wz_{t}=0,\ \
u^{-1}(z_{0}+\theta_{0})+v^{-1}(z_{1}+\theta_{1})+w^{-1}(z_{t}+\theta_{t})=0.$
(132)

We put

$\displaystyle\kappa_{1}:=-(z_{0}+\theta_{0}+z_{1}+\theta_{1}+z_{t}+\theta_{t}),\
\ \kappa_{2}:=z_{0}+z_{1}+z_{t}.$ (133)

Then it holds that $A_{\infty}=\mbox{{\rm diag}}[\kappa_{1},\kappa_{2}]$ and

$\kappa_{1}+\kappa_{2}+\theta_{0}+\theta_{1}+\theta_{t}=0.$ (134)

From the first equation of (132), we find that the $(1,2)$-entry of $A(x,t)$
is of the form

$\frac{p(x)}{x(x-1)(x-t)}$

for some linear polynomial $p(x)$ in $x$. Explicitly, $p(x)$ is expressible as
$p(x)=k(x-y)$ with

$k=(t+1)uz_{0}+tvz_{1}+wz_{t},\ \ \ y=k^{-1}tuz_{0}.$ (135)

We define

$\displaystyle\tilde{z}:=\frac{z_{0}+\theta_{0}}{y}+\frac{z_{1}+\theta_{1}}{y-1}+\frac{z_{t}+\theta_{t}}{y-t},\
\
\tilde{\tilde{z}}:=\tilde{z}-\frac{\theta_{0}}{y}-\frac{\theta_{1}}{y-1}-\frac{\theta_{t}}{y-t}.$
(136)

It is underlined that $\tilde{z}$ and $\tilde{\tilde{z}}$ are given by the
values at $x=y$ of the $(1,1)$-entry and $(2,2)$-entry of $A(x,t)$
respectively, namely $\tilde{z}=A_{11}(y,t)$ and
$\tilde{\tilde{z}}=-A_{22}(y,t)$. From the equations (132),(133),(135),(136),
we obtain the following equalities:

$\displaystyle u=\frac{ky}{tz_{0}},\ \ v=-\frac{k(y-1)}{(t-1)z_{1}},\ \
w=\frac{k(y-t)}{t(t-1)z_{t}},$ (137) $\displaystyle
z_{0}=\frac{y}{t\theta_{\infty}}\Bigl{(}y(y-1)(y-t)\tilde{\tilde{z}}^{2}+\bigl{(}\theta_{1}(y-t)+t\theta_{t}(y-1)$
$\displaystyle\ \ \ \ \ \ \ \
-2\kappa_{2}(y-1)(y-t)\bigr{)}\tilde{\tilde{z}}+\kappa_{2}^{2}(y-t-1)-\kappa_{2}(\theta_{1}+t\theta_{t})\Bigr{)},$
(138) $\displaystyle
z_{1}=-\frac{y-1}{(t-1)\theta_{\infty}}\Bigl{(}y(y-1)(y-t)\tilde{\tilde{z}}^{2}+\bigl{(}(\theta_{1}+\theta_{\infty})(y-t)+t\theta_{t}(y-1)$
$\displaystyle\ \ \ \ \ \ \ \
-2\kappa_{2}(y-1)(y-t)\bigr{)}\tilde{\tilde{z}}+\kappa_{2}^{2}(y-t)-\kappa_{2}(\theta_{1}+t\theta_{t})-\kappa_{1}\kappa_{2}\Bigr{)},$
(139) $\displaystyle
z_{t}=\frac{y-t}{t(t-1)\theta_{\infty}}\Bigl{(}y(y-1)(y-t)\tilde{\tilde{z}}^{2}+\bigl{(}\theta_{1}(y-t)+t(\theta_{t}+\theta_{\infty})(y-1)$
$\displaystyle\ \ \ \ \ \ \ \
-2\kappa_{2}(y-1)(y-t)\bigr{)}\tilde{\tilde{z}}+\kappa_{2}^{2}(y-1)-\kappa_{2}(\theta_{1}+t\theta_{t})-t\kappa_{1}\kappa_{2}\Bigr{)},$
(140)

where we put $\theta_{\infty}:=\kappa_{1}-\kappa_{2}$. These equalities imply
that $\\{u,v,w,z_{0},z_{1},z_{t}\\}$ and thus all the entries of
$A_{0},A_{1},A_{t}$ are expressed in terms of
$\\{y,\tilde{z},k,\theta_{0},\theta_{1},\theta_{t},\kappa_{1},\kappa_{2},t\\}$.
Then, the Schlesinger system (which is nothing but the integrability condition
of (129))

$\displaystyle\frac{dA_{0}}{dt}=\frac{1}{t}[A_{t},A_{0}],\ \
\frac{dA_{1}}{dt}=\frac{1}{t-1}[A_{t},A_{1}],\ \
\frac{dA_{t}}{dt}=\frac{1}{t}[A_{0},A_{t}]+\frac{1}{t-1}[A_{1},A_{t}]$ (141)

is equivalent to the following system of differential equations

$\displaystyle\frac{dy}{dt}=\frac{y(y-1)(y-t)}{t(t-1)}\left(2\tilde{z}-\frac{\theta_{0}}{y}-\frac{\theta_{1}}{y-1}-\frac{\theta_{t}-1}{y-t}\right),$
(142)
$\displaystyle\frac{d\tilde{z}}{dt}=\frac{1}{t(t-1)}\bigl{(}(-3y^{2}+2(1+t)y-t)\tilde{z}^{2}$
$\displaystyle\ \ \ \ \ \ \
+((2y-1-t)\theta_{0}+(2y-t)\theta_{1}+(2y-1)(\theta_{t}-1))\tilde{z}-\kappa_{1}(\kappa_{2}+1)\bigr{)},$
(143) $\displaystyle\frac{d}{dt}\log k=(\theta_{\infty}-1)\frac{y-t}{t(t-1)},$
(144)
$\displaystyle\frac{d\theta_{0}}{dt}=\frac{d\theta_{1}}{dt}=\frac{d\theta_{t}}{dt}=\frac{d\kappa_{1}}{dt}=\frac{d\kappa_{2}}{dt}=0.$
(145)

(We note that, eliminating $\tilde{z}$ from (142) and (143), we obtain the
Painlevé VI equation for $y$ with
$\alpha=\frac{1}{2}(\theta_{\infty}-1)^{2},\beta=-\frac{1}{2}\theta_{0}^{2},\gamma=\frac{1}{2}\theta_{1}^{2},\delta=\frac{1}{2}(1-\theta_{t}^{2})$.)

Now we shall prove Proposition A.1.

###### Proof of Proposition A.1.

First, we start from an Okubo type system in several variables (122) with
$n=3$. By the procedure explained in Appendix B, we obtain a completely
integrable Pfaffian system of rank $2$ replacing
$B_{\infty}=\mbox{diag}[\lambda_{1},\lambda_{2},\lambda_{3}]$ to
$\mbox{diag}[\lambda_{1}-\lambda_{3},\lambda_{2}-\lambda_{3},0]$:

$dZ=\Bigl{(}\Gamma^{(z)}dz+\sum_{i=1}^{3}\Gamma^{(i)}dx_{i}\Bigr{)}Z.$ (146)

Changing the variables $(x_{1},x_{2},x_{3})$ to $(z_{1},z_{2},z_{3})$ and then
setting $x=(z-z_{1})/(z_{2}-z_{1}),t=(z_{3}-z_{1})/(z_{2}-z_{1})$, we find
that (146) is changed into the form of (129) in Fact 2. Therefore we obtain a
solution $y=y(t)$ to the Painlevé VI equation with
$\alpha=\frac{1}{2}(\lambda_{1}-\lambda_{2}-1)^{2},\beta=-\frac{1}{2}(r_{1}-\lambda_{3})^{2},\gamma=\frac{1}{2}(r_{2}-\lambda_{3})^{2},\delta=\frac{1}{2}(1-(r_{3}-\lambda_{3})^{2})$.

Conversely we take a solution $y=y(t)$ to the Painlevé VI equation with
parameters $\\{\theta_{0},\theta_{1},\theta_{t},\theta_{\infty}\\}$. From this
$y$, we can construct $\tilde{z}=\tilde{z}(t)$ and $k=k(t)$ using (142) and
(144) respectively. (We may freely take the integral constant of $k$.) Then,
$\\{y,\tilde{z},k\\}$ constructed in this way satisfy the equations (142),
(143), (144), and thus we obtain a completely integrable Pfaffian system (129)
of rank $2$ using (137), (138), (139), (140). Changing the variables $x,t$ to
$z,z_{1},z_{2},z_{3}$ by
$x=(z-z_{1})/(z_{2}-z_{1}),t=(z_{3}-z_{1})/(z_{2}-z_{1})$ and applying the
procedure in Appendix B, we obtain an Okubo type system in several variables
of rank $3$. ∎

##  Appendix B Okubo type systems of rank $n$ and Pfaffian systems of rank
$n-1$

In this Appendix B, we explain a method of constructing an Okubo type system
in $(n+1)$ variables of rank $n$ from a completely integrable Pfaffian system
of rank $(n-1)$ by using a modification of a technique on middle convolution
developed by Haraoka [12]. For this purpose we first treat an Okubo type
system in $(n+1)$ variables $(z,x_{1},\ldots,x_{n})$ of rank $n$

$dY=\Bigl{(}B^{(z)}dz+\sum_{i=1}^{n}B^{(i)}dx_{i}\Bigr{)}Y,$ (147)

where

$\displaystyle Y=(y_{1},\dots,y_{n})^{t},$ $\displaystyle
B^{(z)}=-(zI_{n}-T)^{-1}B_{\infty},$ $\displaystyle
B^{(i)}=-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty},\ \ \ \ i=1,\dots,n.$

We construct a Pfaffian system of rank $(n-1)$ from (147). From Remark 2.2, we
can change the eigenvalues $\\{\lambda_{1},\dots,\lambda_{n}\\}$ of
$B_{\infty}$ such that one of them is equal to $0$ and, thus, we may suppose
$\lambda_{n}=0$. Then the Okubo type system (147) turns out to be reducible
and we obtain the rank $n-1$ Pfaffian system satisfied by
$Z=(y_{1},\dots,y_{n-1})^{t}$ that consists of the first $(n-1)$ entries of
$Y$:

$dZ=\Bigl{(}\Gamma^{(z)}dz+\sum_{i=1}^{n}\Gamma^{(i)}dx_{i}\Bigr{)}Z,$ (148)

where

$\displaystyle\Gamma^{(z)}=\begin{pmatrix}I_{n-1}&0\end{pmatrix}\bigl{(}-(zI_{n}-T)^{-1}B_{\infty}\bigr{)}\begin{pmatrix}I_{n-1}\\\
0\end{pmatrix},$
$\displaystyle\Gamma^{(i)}=\begin{pmatrix}I_{n-1}&0\end{pmatrix}\bigl{(}-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty}\bigr{)}\begin{pmatrix}I_{n-1}\\\
0\end{pmatrix},\ \ \ i=1,\dots,n.$

The system (148) is also completely integrable.

The aim of this Appendix B is to explain a method of constructing an Okubo
type system of rank $n$ from a system of the form (148). In this Appendix B a
function means a function of $(n+1)$ variables $(x_{1},\ldots,x_{n},z)$
defined on an open set $U\times
U_{z}\subset\mathbb{C}^{n}\times\mathbb{P}^{1}$ otherwise stated, where $z$ is
a non-homogeneous coordinate of $\mathbb{P}^{1}$. We start with an integrable
Pfaffian system of rank $(n-1)$

$dZ=\Bigl{(}\Gamma^{(z)}dz+\sum_{i=1}^{n}\Gamma^{(i)}dx_{i}\Bigr{)}Z$ (149)

which satisfies the following conditions:

(D1) There is a monic polynomial $H(x,z)$ in $z$ of degree $n$ and analytic in
$x$ and $(n-1)\times(n-1)$ matrices
$\Gamma^{(z)},\>\Gamma^{(i)}\>(i=1,\ldots,n)$ such that the entries of
$H(x,z)\Gamma^{(z)},H(x,z)\Gamma^{(i)}$ are polynomials in $z$ and holomorphic
on $U$. In addition, the discriminant
$\delta_{H}=\prod_{i<j}(z_{i}(x)-z_{j}(x))^{2}$ of
$H(x,z)=\prod_{i=1}^{n}(z-z_{i}(x))$ does not identically vanish.

(D2) There are $(n-1)\times(n-1)$ matrices
$\Gamma_{j}^{(z)},\>\Gamma_{j}^{(i)}$ such that the matrices
$\Gamma^{(z)},\>\Gamma^{(i)}\>(i=1,\ldots,n)$ take the forms

$\displaystyle\Gamma^{(z)}=\sum_{j=1}^{n}\frac{\Gamma^{(z)}_{j}}{z-z_{j}(x)},\
\ \ \ \Gamma^{(i)}=\sum_{j=1}^{n}\frac{\Gamma^{(i)}_{j}}{z-z_{j}(x)},\ \
i=1,\dots,n,$

and that each entry of $\Gamma_{j}^{(z)},\>\Gamma_{j}^{(i)}$ is holomorphic on
$W\subset U\setminus\Delta_{H}$, where $\Delta_{H}$ denotes the bifurcation
set of $H$ and $W$ is subject to (W) in Section 2.

(D3) $\mbox{rank}\,\Gamma^{(z)}_{j}=1,\mbox{tr}\Gamma_{j}^{(z)}\neq\pm 1,\
j=1,\ldots,n$.

(D4) $\Gamma_{\infty}=-\sum_{j=1}^{n}\Gamma^{(z)}_{j}$ is a diagonal matrix
whose entries are constants. Moreover put
$\Gamma_{\infty}=\mbox{diag}[\lambda_{1},\ldots,\lambda_{n-1}]$. Then
$\lambda_{i}\not=0\>(i=1,\ldots,n-1)$.

In the sequel, we explain a method how to construct an integrable Pfaffian
system of rank $n$

$dY=\Bigl{(}\hat{\Gamma}^{(z)}dz+\sum_{i=1}^{n}\hat{\Gamma}^{(i)}dx_{i}\Bigr{)}Y$
(150)

which satisfies the following conditions:

(D’1) $\hat{\Gamma}^{(z)},\>\hat{\Gamma}^{(i)}\>(i=1,\ldots,n)$ are $n\times
n$ matrices such that the entries of
$H(x,z)\hat{\Gamma}^{(z)},H(x,z)\hat{\Gamma}^{(i)}$ are polynomials in $z$ and
holomorphic on $U$.

(D’2) There are $n\times n$ matrices
$\hat{\Gamma}_{j}^{(z)},\>\hat{\Gamma}_{j}^{(i)}$ such that the matrices
$\hat{\Gamma}^{(z)},\>\hat{\Gamma}^{(i)}\>(i=1,\ldots,n)$ take the forms

$\displaystyle\hat{\Gamma}^{(z)}=\sum_{j=1}^{n}\frac{\hat{\Gamma}^{(z)}_{j}}{z-z_{j}(x)},\
\ \ \
\hat{\Gamma}^{(i)}=\sum_{j=1}^{n}\frac{\hat{\Gamma}^{(i)}_{j}}{z-z_{j}(x)},\ \
i=1,\dots,n,$

and that each entry of $\hat{\Gamma}_{j}^{(z)},\>\hat{\Gamma}_{j}^{(i)}$ is
holomorphic on $W$.

(D’3) $\mbox{rank}\,\hat{\Gamma}^{(z)}_{j}=1,\ j=1,\ldots,n$.

(D’4) $\hat{\Gamma}_{\infty}=-\sum_{j=1}^{n}\hat{\Gamma}^{(z)}_{j}$ is a
diagonal matrix whose entries are constants.
In virtue of Lemma 2.4, (150) is of Okubo type.

It is underlined here that the singularities of the system (150) coincide with
that of the system (149). The construction of the system (150) from (149) is
accomplished by employing an analytic representation of middle convolution
(cf. [12], [7], [8]).

We are going to construct (150). For a solution $Z$ of (149), let

$\displaystyle\hat{Z}(x_{1},\dots,x_{n},z)=\begin{pmatrix}\hat{Z}_{1}\\\
\vdots\\\ \hat{Z}_{n}\end{pmatrix},$
$\displaystyle\hat{Z}_{j}=\int_{\Delta}\frac{Z(x_{1},\dots,x_{n},t)}{t-z_{j}}(t-z)^{\lambda}\,dt,\
\ \ \ j=1,\dots,n,$

where $\lambda\in\mathbb{C}$ and $\Delta$ denotes a twisted cycle. Then the
following holds.

######  Lemma B.1.

$\hat{Z}$ satisfies the following Pfaffian system of rank $n(n-1)$:

$d\hat{Z}=\Bigl{(}G^{(z)}dz+\sum_{k=1}^{n}G^{(k)}dx_{k}\Bigr{)}\hat{Z},$ (151)

where $G^{(z)},$ $G^{(k)},$ $k=1,\dots,n$ are $n(n-1)\times n(n-1)$-matrices
whose entries, when they are decomposed into blocks of
$(n-1)\times(n-1)$-matrices, are written as follows:

$G^{(z)}=\begin{pmatrix}G^{(z)}_{11}&\cdots&G^{(z)}_{1n}\\\
\vdots&\ddots&\vdots\\\ G^{(z)}_{n1}&\cdots&G^{(z)}_{nn}\end{pmatrix},\ \ \
G^{(k)}=\begin{pmatrix}G^{(k)}_{11}&\cdots&G^{(k)}_{1n}\\\
\vdots&\ddots&\vdots\\\ G^{(k)}_{n1}&\cdots&G^{(k)}_{nn}\end{pmatrix},$
$G^{(z)}_{ij}=\frac{\Gamma^{(z)}_{j}+\lambda\delta_{ij}I_{n-1}}{z-z_{i}},\ \ \
i,j=1,\dots,n,$ (152)

and, for $k=1,\dots,n$,

$\displaystyle G^{(k)}_{ij}=-\frac{\partial z_{i}}{\partial
x_{k}}\frac{\Gamma^{(z)}_{j}}{z-z_{i}}-\bigl{(}\frac{\partial z_{i}}{\partial
x_{k}}-\frac{\partial z_{j}}{\partial
x_{k}}\bigr{)}\frac{\Gamma^{(z)}_{j}}{z_{i}-z_{j}},\ \ \ \ i,j=1,\dots,n,\
i\neq j,$ (153) $\displaystyle G^{(k)}_{ii}=-\frac{\partial z_{i}}{\partial
x_{k}}\frac{\Gamma^{(z)}_{i}+\lambda I_{n-1}}{z-z_{i}}+\sum_{j\neq
i}\bigl{(}\frac{\partial z_{i}}{\partial x_{k}}-\frac{\partial z_{j}}{\partial
x_{k}}\bigr{)}\frac{\Gamma^{(z)}_{j}}{z_{i}-z_{j}},\ \ i=1,\dots,n.$ (154)

###### Proof.

First, we show (152). In virtue of the definition of $\hat{Z}_{j}$,

$\displaystyle(z-z_{j})\frac{\partial\hat{Z}_{j}}{\partial z}$
$\displaystyle=-\lambda\int_{\Delta}(z-z_{j})\frac{Z(t)}{t-z_{j}}(t-z)^{\lambda-1}dt$
$\displaystyle=\lambda\hat{Z}_{j}-\lambda\int_{\Delta}Z(t)(t-z)^{\lambda-1}dt$
$\displaystyle=\lambda\hat{Z}_{j}+\sum_{k=1}^{n}\int_{\Delta}\frac{\Gamma^{(z)}_{k}}{t-z_{k}}Z(t)(t-z)^{\lambda}dt$
$\displaystyle=\lambda\hat{Z}_{j}+\sum_{k=1}^{n}\Gamma^{(z)}_{k}\hat{Z}_{k},$

where we have employed that

$\displaystyle 0=\int_{\Delta}d\bigl{(}Z(t)(t-z)^{\lambda}\bigr{)}=$
$\displaystyle\int_{\Delta}\frac{\partial Z}{\partial
t}(t)(t-z)^{\lambda}dt+\lambda\int_{\Delta}Z(t)(t-z)^{\lambda-1}dt$

by Stokes’ theorem.

Next, we show (153) and (154). With the help of Stokes’ theorem, we have

$\displaystyle
0=\int_{\Delta}d\left(\frac{Z(t)}{(t-z_{j})}(t-z)^{\lambda}\right)=$
$\displaystyle-\int_{\Delta}\frac{Z(t)}{(t-z_{j})^{2}}(t-z)^{\lambda}+\lambda\int_{\Delta}\frac{Z(t)}{t-z_{j}}(t-z)^{\lambda-1}dt$
$\displaystyle\ \
+\sum_{k=1}^{n}\Gamma^{(z)}_{k}\int_{\Delta}\frac{Z(t)}{(t-z_{j})(t-z_{k})}(t-z)^{\lambda}dt,$

and, thus, it holds that

$(I_{n-1}-\Gamma^{(z)}_{j})\int_{\Delta}\frac{Z(t)}{(t-z_{j})^{2}}(t-z)^{\lambda}dt=-\frac{\partial\hat{Z}_{j}}{\partial
z}+\sum_{k\neq
j}\frac{\Gamma^{(z)}_{k}}{z_{j}-z_{k}}(\hat{Z}_{j}-\hat{Z}_{k}).$

Then we have

$\displaystyle\frac{\partial\hat{Z}_{i}}{\partial x_{k}}$
$\displaystyle=\int_{\Delta}\frac{\frac{\partial Z}{\partial
x_{k}}(t)}{t-z_{i}}(t-z)^{\lambda}dt+\frac{\partial z_{i}}{\partial
x_{k}}\int_{\Delta}\frac{Z(t)}{(t-z_{i})^{2}}(t-z)^{\lambda}dt$
$\displaystyle=\int_{\Delta}\frac{1}{t-z_{i}}\Bigl{(}\sum_{j=1}^{n}\frac{\Gamma^{(k)}_{i}}{t-z_{j}}\Bigr{)}Z(t)(t-z)^{\lambda}dt+\frac{\partial
z_{i}}{\partial
x_{k}}\int_{\Delta}\frac{Z(t)}{(t-z_{i})^{2}}(t-z)^{\lambda}dt$
$\displaystyle=\sum_{j\neq
i}\frac{\Gamma^{(k)}_{j}}{z_{i}-z_{j}}(\hat{Z}_{i}-\hat{Z}_{j})+\Bigl{(}\Gamma^{(k)}_{i}+\frac{\partial
z_{i}}{\partial
x_{k}}\Bigr{)}\int_{\Delta}\frac{Z(t)}{(t-z_{i})^{2}}(t-z)^{\lambda}dt$
$\displaystyle=-\sum_{j\neq i}\frac{\frac{\partial z_{j}}{\partial
x_{k}}}{z_{i}-z_{j}}\Gamma^{(z)}_{j}(\hat{Z}_{i}-\hat{Z}_{j})+\frac{\partial
z_{i}}{\partial x_{k}}\Bigl{(}-\frac{\partial\hat{Z}_{i}}{\partial
z}+\sum_{j\neq
i}\frac{\Gamma^{(z)}_{j}}{z_{i}-z_{j}}(\hat{Z}_{i}-\hat{Z}_{j})\Bigr{)}$
$\displaystyle=-\frac{\partial z_{i}}{\partial
x_{k}}\Bigl{(}\frac{\lambda}{z-z_{i}}\hat{Z}_{i}+\sum_{j=1}^{n}\frac{\Gamma^{(z)}_{j}}{z-z_{i}}\hat{Z}_{j}\Bigr{)}+\sum_{j\neq
i}\Bigl{(}\frac{\partial z_{i}}{\partial x_{k}}-\frac{\partial z_{j}}{\partial
x_{k}}\Bigr{)}\frac{\Gamma^{(z)}_{j}}{z_{i}-z_{j}}(\hat{Z}_{i}-\hat{Z}_{j}).$

∎

######  Lemma B.2.

Let

$\mathcal{K}=\left\\{\left.\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{n}\end{pmatrix}\right|\,v_{i}\in\ker\Gamma^{(z)}_{i}\
(i=1,\dots,n)\right\\}\subset\mathcal{M}_{W}^{n(n-1)},$
$\mathcal{L}=\ker\begin{pmatrix}\Gamma^{(z)}_{1}+\lambda
I_{n-1}&\Gamma^{(z)}_{2}&\cdots&\Gamma^{(z)}_{n}\\\
\Gamma^{(z)}_{1}&\Gamma^{(z)}_{2}+\lambda I_{n-1}&\cdots&\Gamma^{(z)}_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
\Gamma^{(z)}_{1}&\Gamma^{(z)}_{2}&\cdots&\Gamma^{(z)}_{n}+\lambda
I_{n-1}\end{pmatrix}\subset\mathcal{M}_{W}^{n(n-1)},$

where $\mathcal{M}_{W}$ denotes the field of meromorphic functions on $W$.
Then

$\displaystyle\Bigl{(}\frac{\partial}{\partial
z}-G^{(z)}\Bigr{)}\mathcal{K}\subset\mathcal{K},\ \
\Bigl{(}\frac{\partial}{\partial
z}-G^{(z)}\Bigr{)}\mathcal{L}\subset\mathcal{L},$ (155)
$\displaystyle\Bigl{(}\frac{\partial}{\partial
x_{i}}-G^{(i)}\Bigr{)}\mathcal{K}\subset\mathcal{K},\ \
\Bigl{(}\frac{\partial}{\partial
x_{i}}-G^{(i)}\Bigr{)}\mathcal{L}\subset\mathcal{L},\ \ \ \ i=1,\dots,n.$
(156)

###### Proof.

From the integrability condition of (147), we have

$\displaystyle\Gamma^{(i)}_{j}=-\frac{\partial z_{j}}{\partial
x_{i}}\Gamma^{(z)}_{j},\ \ i,j=1,\dots,n,$ (157)
$\displaystyle\frac{\partial\Gamma^{(z)}_{j}}{\partial x_{i}}=\sum_{k\neq
j}\frac{[\Gamma^{(i)}_{j},\Gamma^{(z)}_{k}]+[\Gamma^{(i)}_{k},\Gamma^{(z)}_{j}]}{z_{j}-z_{k}},\
\ i,j=1,\dots,n.$ (158)

We shall prove $(\frac{\partial}{\partial
z}-G^{(z)})\mathcal{K}\subset\mathcal{K}$ and $(\frac{\partial}{\partial
x_{k}}-G^{(k)})\mathcal{K}\subset\mathcal{K}$. By definition,

$v=\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{n}\end{pmatrix}\in\mathcal{K}\Longleftrightarrow\Gamma^{(z)}_{j}v_{j}=0,\ \
j=1,\dots,n,$

and, thus, we will show

$\Gamma^{(z)}_{i}\left(\frac{\partial v_{i}}{\partial
z}-\sum_{j=1}^{n}G^{(z)}_{ij}v_{j}\right)=0,\ \
\Gamma^{(z)}_{i}\left(\frac{\partial v_{i}}{\partial
x_{k}}-\sum_{j=1}^{n}G^{(k)}_{ij}v_{j}\right)=0,\ \ \ \ i,k=1,\dots,n$

for any $v\in\mathcal{K}$. The former is obvious from

$\frac{\partial v_{i}}{\partial z}=0,\ \ \ \
\sum_{j=1}^{n}G^{(z)}_{ij}v_{j}=\frac{\lambda}{z-z_{i}}v_{i}.$

For the latter, differentiating the both sides of $\Gamma^{(z)}_{i}v_{i}=0$
with respect to $x_{k}$, we have

$\Gamma^{(z)}_{i}\frac{\partial v_{i}}{\partial
x_{k}}=-\frac{\partial\Gamma^{(z)}_{i}}{\partial x_{k}}v_{i},$

and, by (158), we have

$-\frac{\partial\Gamma^{(z)}_{i}}{\partial x_{k}}v_{i}=\sum_{j\neq
i}\frac{\frac{\partial z_{i}}{\partial x_{k}}-\frac{\partial z_{j}}{\partial
x_{k}}}{z_{i}-z_{j}}\Gamma^{(z)}_{i}\Gamma^{(z)}_{j}v_{i}.$

In turn, it holds that

$\sum_{j=1}^{n}G^{(k)}_{ij}v_{j}=-\frac{\partial z_{i}}{\partial
x_{k}}\frac{\lambda}{z-z_{i}}v_{i}+\sum_{j\neq i}\frac{\frac{\partial
z_{i}}{\partial x_{k}}-\frac{\partial z_{j}}{\partial
x_{k}}}{z_{i}-z_{j}}\Gamma^{(z)}_{j}v_{i}.$

Hence we have

$\Gamma^{(z)}_{i}\left(\frac{\partial v_{i}}{\partial
x_{k}}-\sum_{j=1}^{n}G^{(k)}_{ij}v_{j}\right)=\frac{\partial z_{i}}{\partial
x_{k}}\frac{\lambda}{z-z_{i}}\Gamma^{(z)}_{i}v_{i}=0.$

It is possible to prove $(\frac{\partial}{\partial
z}-G^{(z)})\mathcal{L}\subset\mathcal{L}$ and $(\frac{\partial}{\partial
x_{k}}-G^{(k)})\mathcal{L}\subset\mathcal{L}$ similarly. However we omit the
details. ∎

By Lemma B.2, we can induce a Pfaffian system on
$\mathcal{M}_{W}^{n(n-1)}/(\mathcal{K}+\mathcal{L})$ from (151). Noting that
$\mbox{rank}\,\Gamma^{(z)}_{j}=1$ $(j=1,\dots,n)$, we find that
$\Gamma^{(z)}_{j}$ is written as follows:

$\Gamma^{(z)}_{j}=-\begin{pmatrix}b_{1j}\\\ \vdots\\\
b_{n-1,j}\end{pmatrix}\begin{pmatrix}a_{j1}&\cdots&a_{j,n-1}\end{pmatrix}\Gamma_{\infty}.$

The equality $\Gamma_{\infty}=-\sum_{j=1}^{n}\Gamma^{(z)}_{j}$ implies

$\begin{pmatrix}b_{11}&\cdots&b_{1n}\\\ \vdots&\ddots&\vdots\\\
b_{n-1,1}&\cdots&b_{n-1,n}\end{pmatrix}\begin{pmatrix}a_{11}&\cdots&a_{1,n-1}\\\
\vdots&\ddots&\vdots\\\ a_{n,1}&\cdots&a_{n,n-1}\end{pmatrix}=I_{n-1}.$

At this stage, we choose vectors
$\begin{pmatrix}b_{n,1}&\cdots&b_{n,n}\end{pmatrix}$ and
$\begin{pmatrix}a_{1,n}&\cdots&a_{n,n}\end{pmatrix}^{t}$ so that

$\begin{pmatrix}b_{11}&\cdots&b_{1n}\\\ \vdots&\ddots&\vdots\\\
b_{n,1}&\cdots&b_{n,n}\end{pmatrix}\begin{pmatrix}a_{11}&\cdots&a_{1,n}\\\
\vdots&\ddots&\vdots\\\ a_{n,1}&\cdots&a_{n,n}\end{pmatrix}=I_{n}.$

We put

$P=\begin{pmatrix}b_{11}&\cdots&b_{1n}\\\ \vdots&\ddots&\vdots\\\
b_{n1}&\cdots&b_{nn}\end{pmatrix}.$

Then it follows from the definition that

$P^{-1}=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\\ \vdots&\ddots&\vdots\\\
a_{n1}&\cdots&a_{nn}\end{pmatrix}.$

######  Lemma B.3.

Take a basis $\\{k_{1},\dots,k_{n(n-2)}\\}$ of $\mathcal{K}$. Let ${\bf
e}_{i}\in\mathbb{C}^{n-1}$ $(i=1,\dots,n-1)$ be the $i$-th fundamental vector
of $\mathbb{C}^{n-1}$ and let

$l_{i}=-\lambda_{i}^{-1}\begin{pmatrix}{\bf e}_{i}\\\ \vdots\\\ {\bf
e}_{i}\end{pmatrix}\in\mathbb{C}^{n(n-1)}\subset\mathcal{M}_{W}^{n(n-1)},\ \
i=1,\dots,n-1.$

For each $i$ $(i=1,\dots,n)$, we can choose $j_{i}\in\\{1,\dots,n-1\\}$ such
that $\prod_{i=1}^{n}a_{i,j_{i}}\neq 0$. For simplicity, we assume
$\prod_{i=1}^{n}a_{i1}\neq 0$ (i.e. $j_{i}=1$ for any $i$). Put

$l_{n}=-\lambda_{1}^{-1}\begin{pmatrix}a_{1n}a_{11}^{-1}{\bf e}_{1}\\\
a_{2n}a_{21}^{-1}{\bf e}_{1}\\\ \vdots\\\ a_{nn}a_{n1}^{-1}{\bf
e}_{1}\end{pmatrix}\in\mathcal{M}_{W}^{n(n-1)}.$

Then $\\{k_{1},\dots,k_{n(n-2)},l_{1},\dots,l_{n}\\}$ forms a basis of
$\mathcal{M}_{W}^{n(n-1)}$. If
$\lambda\not\in\\{\lambda_{1},\dots,\lambda_{n-1},0\\}$, then
$\mathcal{L}=\\{0\\}$ holds.

###### Proof.

It can be proved straightforwardly. ∎

In the sequel, we assume
$\lambda\not\in\\{\lambda_{1},\dots,\lambda_{n-1},0\\}$. Then we can take
$\\{l_{1},\dots,l_{n}\\}$ as a basis of
$\mathcal{M}_{W}^{n(n-1)}/\mathcal{K}$. We give the representation matrix of
the middle convolution of (149) with respect to the basis
$\\{l_{1},\dots,l_{n}\\}$. We define an $n(n-1)\times n(n-1)$ matrix $K$ by

$K=(k_{1},\dots,k_{n(n-2)},l_{1},\dots,l_{n}).$

Changing the unknown functions by $\hat{Z}=K\hat{Y}$, the Pfaffian system
(151) is changed to

$d\hat{Y}=\Bigl{(}K^{-1}G^{(z)}Kdz+\sum_{k=1}^{n}K^{-1}G^{(k)}Kdx_{k}-K^{-1}dK\Bigr{)}\hat{Y}.$

From Lemma B.2, there exist $n\times n$ matrices
$\hat{\Gamma}^{(z)^{\prime}}$, $\hat{\Gamma}^{(i)^{\prime}}$ such that

$K^{-1}\Bigl{(}G^{(z)}-\frac{\partial}{\partial
z}\Bigr{)}K=\begin{pmatrix}*&*\\\ O&\hat{\Gamma}^{(z)^{\prime}}\end{pmatrix},\
\ \ \ K^{-1}\Bigl{(}G^{(i)}-\frac{\partial}{\partial
x_{i}}\Bigr{)}K=\begin{pmatrix}*&*\\\
O&\hat{\Gamma}^{(i)^{\prime}}\end{pmatrix}.$

They are written as

$\hat{\Gamma}^{(i)^{\prime}}=(\underbrace{O,I_{n}}_{n(n-1)})K^{-1}\Bigl{(}G^{(i)}-\frac{\partial}{\partial
x_{i}}\Bigr{)}(l_{1},\dots,l_{n}),$

where we have

$\displaystyle(O,I_{n})K^{-1}$ $\displaystyle=$
$\displaystyle-\left(\begin{pmatrix}b_{11}\\\ \vdots\\\
b_{n1}\end{pmatrix}\begin{pmatrix}a_{11}&\cdots&a_{1,n-1}\end{pmatrix}\Gamma_{\infty},\cdots,\begin{pmatrix}b_{1n}\\\
\vdots\\\
b_{nn}\end{pmatrix}\begin{pmatrix}a_{n1}&\cdots&a_{n,n-1}\end{pmatrix}\Gamma_{\infty}\right).$

Besides, we find

$G^{(z)}(l_{1},\dots,l_{n})=-\begin{pmatrix}\displaystyle\frac{I_{n-1}}{z-z_{1}}&{}&O\\\
{}&\ddots&{}\\\
O&{}&\displaystyle\frac{I_{n-1}}{z-z_{n}}\end{pmatrix}(l_{1},\dots,l_{n})\hat{\Gamma}_{\infty},$

and for $i=1,\dots,n-1$, we find

$\displaystyle G^{(i)}(l_{1},\dots,l_{n})=$ $\displaystyle\
\begin{pmatrix}\displaystyle\frac{\partial z_{1}}{\partial
x_{i}}\frac{I_{n-1}}{z-z_{1}}&{}&O\\\ {}&\ddots&{}\\\
O&{}&\displaystyle\frac{\partial z_{n}}{\partial
x_{i}}\frac{I_{n-1}}{z-z_{n}}\end{pmatrix}(l_{1},\dots,l_{n})\hat{\Gamma}_{\infty}$
$\displaystyle\ \ +\begin{pmatrix}O&\displaystyle\sum_{j\neq
1}(a_{1n}a_{11}^{-1}-a_{jn}a_{j1}^{-1})\displaystyle\Bigl{(}\frac{\partial
z_{1}}{\partial x_{i}}-\frac{\partial z_{j}}{\partial
x_{i}}\Bigr{)}\frac{a_{j1}{\bf b}_{j}}{z_{1}-z_{j}}\\\ \vdots&\vdots\\\
O&\displaystyle\sum_{j\neq
n}(a_{nn}a_{n1}^{-1}-a_{jn}a_{j1}^{-1})\displaystyle\Bigl{(}\frac{\partial
z_{n}}{\partial x_{i}}-\frac{\partial z_{j}}{\partial
x_{i}}\Bigr{)}\frac{a_{j1}{\bf b}_{j}}{z_{n}-z_{j}}\end{pmatrix},$

where

$\hat{\Gamma}_{\infty}=\begin{pmatrix}\lambda_{1}-\lambda&{}&{}&{}\\\
{}&\ddots&{}&{}\\\ {}&{}&\lambda_{n-1}-\lambda&{}\\\
{}&{}&{}&-\lambda\end{pmatrix},$

and ${\bf b}_{j}=\begin{pmatrix}b_{1j}&\dots&b_{n-1,j}\end{pmatrix}^{t},$
$j=1,\dots,n.$

Consequently, we obtain

$\displaystyle\hat{\Gamma}^{(z)^{\prime}}=-\sum_{j=1}^{n}\frac{1}{z-z_{j}}\begin{pmatrix}b_{1j}\\\
\vdots\\\
b_{nj}\end{pmatrix}\begin{pmatrix}a_{j1}&\cdots&a_{jn}\end{pmatrix}\hat{\Gamma}_{\infty},$
$\displaystyle\hat{\Gamma}^{(i)^{\prime}}=\sum_{j=1}^{n}\frac{\frac{\partial
z_{j}}{\partial x_{i}}}{z-z_{j}}\begin{pmatrix}b_{1j}\\\ \vdots\\\
b_{nj}\end{pmatrix}\begin{pmatrix}a_{j1}&\cdots&a_{jn}\end{pmatrix}\hat{\Gamma}_{\infty}+\begin{pmatrix}O&{\bf
0}\\\ {\bf 0}&c_{i}\end{pmatrix},\ \ \ i=1,\dots,n,$

where $\begin{pmatrix}O&{\bf 0}\\\ {\bf 0}&c_{i}\end{pmatrix}$ corresponds to
“$\mathcal{E}$-part” of $\hat{\Gamma}^{(i)^{\prime}}$ (see Lemma 2.1 for
$\mathcal{E}$-part) and therefore $c_{i}$ is written as
$c_{i}=\epsilon^{-1}\frac{\partial\epsilon}{\partial x_{i}}$ in terms of some
$\epsilon\in\mathcal{O}^{\times}_{U}$.

Finally, we can conclude that the matrices
$\hat{\Gamma}^{(z)},\hat{\Gamma}^{(i)}$ appeared in the system (150) are given
by

$\displaystyle\hat{\Gamma}^{(z)}$
$\displaystyle=\mbox{diag}(1,\dots,1,\epsilon^{-1})\,\hat{\Gamma}^{(z)^{\prime}}\mbox{diag}(1,\dots,1,\epsilon)=-\sum_{j=1}^{n}\frac{1}{z-z_{j}}\begin{pmatrix}b_{1j}\\\
\vdots\\\
\epsilon^{-1}b_{nj}\end{pmatrix}\begin{pmatrix}a_{j1}&\cdots&\epsilon
a_{jn}\end{pmatrix}\hat{\Gamma}_{\infty},$ $\displaystyle\hat{\Gamma}^{(i)}$
$\displaystyle=\mbox{diag}(1,\dots,1,\epsilon^{-1})\,\hat{\Gamma}^{(i)^{\prime}}\mbox{diag}(1,\dots,1,\epsilon)-\begin{pmatrix}O&{\bf
0}\\\ {\bf 0}&\epsilon^{-1}\frac{\partial\epsilon}{\partial
x_{i}}\end{pmatrix}$ $\displaystyle=\sum_{j=1}^{n}\frac{\frac{\partial
z_{j}}{\partial x_{i}}}{z-z_{j}}\begin{pmatrix}b_{1j}\\\ \vdots\\\
\epsilon^{-1}b_{nj}\end{pmatrix}\begin{pmatrix}a_{j1}&\cdots&\epsilon
a_{jn}\end{pmatrix}\hat{\Gamma}_{\infty},\ \ \ i=1,\dots,n.$

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Mitsuo Kato
---
Department of Mathematics, Colledge of Educations
University of the Ryukyus
Nishihara-cho, Okinawa 903-0213 JAPAN
Toshiyuki Mano
Department of Mathematical Sciences, Faculty of Sciences
University of the Ryukyus
Nishihara-cho, Okinawa 903-0213 JAPAN
Jiro Sekiguchi
Department of Mathematics, Faculty of Engineering
Tokyo University of Agriculture and Technology
Koganei, Tokyo 184-8588 JAPAN