The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces

The Third, Fifth and Sixth Painlevé Equations
on Weighted Projective Spaces

Hayato CHIBA

H. Chiba

Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395,
Japan<EMAIL_ADDRESS>
Received September 17, 2015, in final form February 18, 2016; Published online
February 23, 2016

The third, fifth and sixth Painlevé equations are studied by means of the
weighted projective spaces $\mathbb{C}P^{3}(p,q,r,s)$ with suitable weights
$(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular
normal forms of the equations, symplectic atlases of the spaces of initial
conditions, Riccati solutions and Boutroux’s coordinates are systematically
studied in a unified way with the aid of the orbifold structure of
$\mathbb{C}P^{3}(p,q,r,s)$ and dynamical systems theory.

Painlevé equations; weighted projective space

34M35; 34M45; 34M55

##  1 Introduction

The first to sixth Painlevé equations written in Hamiltonian forms are given
by

$\displaystyle(\text{P}_{J})\colon\ \frac{dx}{dz}=-\frac{\partial
H_{J}}{\partial y},\qquad\frac{dy}{dz}=\frac{\partial H_{J}}{\partial x},$

${J}=\text{I},\text{II},\text{IV},\text{III}(D_{8}),\text{III}(D_{7}),\text{III}(D_{6}),\text{V},\text{VI}$,
with the Hamiltonian functions defined as

$\displaystyle H_{\text{I}}=\frac{1}{2}x^{2}-2y^{3}-zy,$ $\displaystyle
H_{\text{II}}=\frac{1}{2}x^{2}-\frac{1}{2}y^{4}-\frac{1}{2}zy^{2}-\alpha y,$
$\displaystyle H_{\text{IV}}=-xy^{2}+x^{2}y-2xyz-2\alpha x+2\beta y,$
$\displaystyle zH_{\text{III}(D_{8})}=x^{2}y^{2}-\frac{z}{2y}-\frac{1}{2}y,$
$\displaystyle zH_{\text{III}(D_{7})}=x^{2}y^{2}+zx+y+\alpha xy,$
$\displaystyle
zH_{\text{III}(D_{6})}=x^{2}y^{2}-xy^{2}+zx+(\alpha+\beta)xy-\alpha y,$
$\displaystyle
zH_{\text{V}}=x(x+z)y(y-1)+\alpha_{2}yz-\alpha_{3}xy-\alpha_{1}x(y-1),$
$\displaystyle
z(z-1)H_{\text{VI}}=y(y-1)(y-z)x^{2}+\alpha_{2}(\alpha_{1}+\alpha_{2})(y-z)$
$\displaystyle\hphantom{z(z-1)H_{\text{VI}}=}{}-\left(\alpha_{4}(y-1)(y-z)+\alpha_{3}y(y-z)+\alpha_{0}y(y-1)\right)x,$

where $\alpha_{i},\beta\in\mathbb{C}$ are arbitrary parameters. Parameters of
$H_{\text{VI}}$ satisfy the constraint
$\alpha_{0}+\alpha_{1}+2\alpha_{2}+\alpha_{3}+\alpha_{4}=0$. The third
Painlevé equation is divided into three cases $\text{III}(D_{8})$,
$\text{III}(D_{7})$, $\text{III}(D_{6})$ due to the geometry of the spaces of
initial conditions [7, 8]. See [9] for the list of Hamiltonians and Bäcklund
transformations written in these coordinates, although $H_{\text{III}(D_{8})}$
here is obtained by putting $x\mapsto x-1/(2y)$ for that in [9].

Among these Hamiltonians, only $(\text{P}_{\text{I}})$,
$(\text{P}_{\text{II}})$ and $(\text{P}_{\text{IV}})$ are polynomials with
respect to both of the dependent variables $x$, $y$ and the independent
variable $z$. In [2, 3], $(\text{P}_{\text{I}})$, $(\text{P}_{\text{II}})$ and
$(\text{P}_{\text{IV}})$ are studied by means of a weighted projective space
$\mathbb{C}P^{3}(p,q,r,s)$, whose weight $(p,q,r,s)$ is one of the invariants
of the equation determined by the Newton polyhedron. In particular, the
Painlevé property, the spaces of initial conditions and Kovalevskaya exponents
are investigated in detail. The purpose in this paper is to extend the
previous result to the third, fifth and sixth Painlevé equations, whose Newton
polyhedrons are degenerate.

According to [2, 3], let us recall the definition of the Newton diagram of a
polynomial differential system. Consider the system of polynomial differential
equations

$\displaystyle\frac{dx_{i}}{dz}=f_{i}(x_{1},\dots,x_{m},z),\qquad
i=1,\dots,m.$ (1.1)

The exponent of a monomial $x_{1}^{\mu_{1}}\cdots x_{m}^{\mu_{m}}z^{\eta}$
included in the right hand side $f_{i}$ is defined as
$(\mu_{1},\dots,\mu_{i-1},\mu_{i}-1,\mu_{i+1},\dots,\mu_{m},\eta+1)$. Each
exponent specifies a point of the integer lattice in $\mathbb{R}^{m+1}$. The
Newton polyhedron of (1.1) is the convex hull of the union of the positive
quadrants $\mathbb{R}_{+}^{m+1}$ with vertices at the exponents of the
monomials which appear in the system. The Newton diagram of the system is the
union of the compact faces of its Newton polyhedron.

We also consider the perturbative system

$\displaystyle\frac{dx_{i}}{dz}=f_{i}(x_{1},\dots,x_{m},z)+g_{i}(x_{1},\dots,x_{m},z),\qquad
i=1,\dots,m,$ (1.2)

where $f_{i}$ and $g_{i}$ are polynomials. We suppose that

  1. (A1)

the Newton polyhedron of the truncated system (1.1) has only one compact face
and all exponents of monomials included in (1.1) lie on the face.

In this case, there is a tuple of relatively prime positive integers
$(p_{1},\dots,p_{m},r,s)$ and a hyperplane $p_{1}x_{1}+\cdots+p_{m}x_{m}+rz=s$
in $\mathbb{R}^{m+1}$ such that all exponents lie on the plane; i.e., any
monomials $x_{1}^{\mu_{1}}\cdots x_{m}^{\mu_{m}}z^{\eta}$ included in $f_{i}$
satisfy

$\displaystyle
p_{1}\mu_{1}+\cdots+p_{i}(\mu_{i}-1)+\cdots+p_{m}\mu_{m}+r(\eta+1)=s.$

In [2], we further suppose that $s-r=1$ though it is not so essential. To
regard $g_{i}$ as a perturbation, we suppose that

  1. (A2)

any monomials $x_{1}^{\mu_{1}}\cdots x_{m}^{\mu_{m}}z^{\eta}$ included in
$g_{i}$ satisfy

$\displaystyle
p_{1}\mu_{1}+\cdots+p_{i}(\mu_{i}-1)+\cdots+p_{m}\mu_{m}+r(\eta+1)<s$

(this implies that the exponents of $g_{i}$ lie on the lower side of the
hyperplane).

Due to the property of the Newton diagram, it is easy to verify that the
truncated system (1.1) is invariant under the $\mathbb{Z}_{s}$-action given by

$\displaystyle(x_{1},\dots,x_{m},z)\mapsto\big{(}\omega^{p_{1}}x_{1},\dots,\omega^{p_{m}}x_{m},\omega^{r}z\big{)},\qquad\omega:=e^{2\pi
i/s}.$ (1.3)

We require the same for equation (1.2):

  1. (A3)

Equation (1.2) is invariant under the $\mathbb{Z}_{s}$-action (1.3).

A tuple of positive integers $(p_{1},\dots,p_{m},r,s)$ is called the weight of
the system (1.2). It is known that there is a one-to-one correspondence
between nondegenerate Newton diagrams and toric varieties. If exponents lie on
the unique plane $p_{1}x_{1}+\cdots+p_{m}x_{m}+rx=s$ (assumption (A1)), then
the associated toric variety is the weighted projective space
$\mathbb{C}P^{m+1}(p_{1},\dots,p_{m}r,s)$, which is an $(m+1)$-dimensional
orbifold, see Section 2.1 for the definition.

The first, second and fourth Painlevé equations satisfy the above assumptions.
For the first Painlevé equation $x^{\prime}=6y^{2}+z$, $y^{\prime}=x$, put
$f=(6y^{2}+z,x)$ and $g=(0,0)$. The Newton diagram is determined by three
points $(-1,2,1)$, $(-1,0,2)$ and $(1,-1,1)$. They lie on the unique plane
$3x+2y+4z=5$. For the second Painlevé equation $x^{\prime}=2y^{3}+yz+\alpha$,
$y^{\prime}=x$ with a parameter $\alpha$, put $f=(2y^{3}+yz,x)$ and
$g=(\alpha,0)$. The Newton diagram is determined by points $(-1,3,1)$,
$(-1,1,2)$, $(1,-1,1)$, which lie on the plane $2x+y+2z=3$. For the fourth
Painlevé equation, put $f=(-x^{2}+2xy+2xz,-y^{2}+2xy-2yz)$ and
$g=(-2\beta,-2\alpha)$. The Newton diagram is given by the unique face on the
plane $x+y+z=2$ passing through the exponents $(1,0,1)$, $(0,1,1)$ and
$(0,0,2)$. Hence, the weighted projective spaces associated with them are
given by $\mathbb{C}P^{3}(3,2,4,5)$, $\mathbb{C}P^{3}(2,1,2,3)$ and
$\mathbb{C}P^{3}(1,1,1,2)$, respectively. Note that $g$ consists of terms
including arbitrary parameters $\alpha$, $\beta$ for these systems.

The orbifold $\mathbb{C}P^{m+1}(p_{1},\dots,p_{m},r,s)$ is regarded as a
compactification of the phase space
$\mathbb{C}^{m+1}=\\{(x_{1},\dots,x_{m},z)\\}$ of the system (1.2). In [2, 3],
the system (1.2), in particular the first, second and fourth Painlevé
equations, are studied with the aid of the geometry of
$\mathbb{C}P^{m+1}(p_{1},\dots$, $p_{m},r,s)$. In this paper, the third, fifth
and sixth Painlevé equations will be investigated, for which the Newton
polyhedrons are degenerate and do not satisfy (A1).

The third Painlevé equation of type $D_{6}$ is explicitly given by

$\displaystyle(\text{P}_{\text{III}(D_{6})})\colon\
\begin{cases}zx^{\prime}=-2x^{2}y+2xy-(\alpha+\beta)x+\alpha,\\\
zy^{\prime}=2xy^{2}-y^{2}+z+(\alpha+\beta)y.\end{cases}$ (1.4)

Put $f=(-2x^{2}y+2xy,2xy^{2}-y^{2}+z)$ and
$g=(-(\alpha+\beta)x+\alpha,(\alpha+\beta)y)$. Exponents of $f$ are given by
$(1,1,0)$, $(0,1,0)$, $(0,-1,1)$. The Newton polyhedron generated by them has
no compact faces because the positive quadrant $\mathbb{R}_{+}^{3}$ with a
vertex at $(0,1,0)$ includes $(1,1,0)$; the Newton diagram is empty.
Nevertheless, these three points lie on the plane $y+2z=1$. Hence, we define
the weight of $(\text{P}_{\text{III}(D_{6})})$ by $(0,1,2,1)$ and use the
weighted projective space $\mathbb{C}P^{3}(0,1,2,1)$, which is not compact
because of the nonpositive weight.

The third Painlevé equation of type $D_{7}$ is given by

$\displaystyle(\text{P}_{\text{III}(D_{7})})\colon\
\begin{cases}zx^{\prime}=-2x^{2}y-1-\alpha x,\\\ zy^{\prime}=2xy^{2}+z+\alpha
y.\end{cases}$

Put $f=(-2x^{2}y-1,2xy^{2}+z)$ and $g=(-\alpha x,\alpha y)$. Exponents of $f$
are given by $(1,1,0),(-1,0,0)$ and $(0,-1,1)$. The Newton diagram is again
empty, however, these three points lie on the plane $-x+2y+3z=1$. Thus we
define the weight of $(\text{P}_{\text{III}(D_{7})})$ by $(-1,2,3,1)$ and use
the weighted projective space $\mathbb{C}P^{3}(-1,2,3,1)$.

The third Painlevé equation of type $D_{8}$ is given by

$\displaystyle(\text{P}_{\text{III}(D_{8})})\colon\ \begin{cases}\displaystyle
zx^{\prime}=\frac{1}{2}-2x^{2}y-\frac{z}{2y^{2}},\\\
zy^{\prime}=2xy^{2}.\end{cases}$

There are no parameters and put $g=(0,0)$. Exponents of the system are
$(1,1,0)$, $(-1,0,0)$ and $(-1,-2,1)$. The Newton polyhedron is degenerate,
however, these three points lie on the plane $-x+2y+4z=1$. Thus we define the
weight of $(\text{P}_{\text{III}(D_{8})})$ to be $(-1,2,4,1)$ and use the
weighted projective space $\mathbb{C}P^{3}(-1,2,4,1)$.

The fifth Painlevé equation is given by

$\displaystyle(\text{P}_{\text{V}})\colon\
\begin{cases}zx^{\prime}=-2x^{2}y+x^{2}+xz-2xyz+(\alpha_{1}+\alpha_{3})x-\alpha_{2}z,\\\
zy^{\prime}=2xy^{2}-2xy-
yz+y^{2}z-(\alpha_{1}+\alpha_{3})y+\alpha_{1}.\end{cases}$

Put $f=(-2x^{2}y+x^{2}+xz-2xyz,2xy^{2}-2xy-yz+y^{2}z)$ and
$g=((\alpha_{1}+\alpha_{3})x-\alpha_{2}z,-(\alpha_{1}+\alpha_{3})y+\alpha_{1})$.
Exponents of $f$ are $(1,1,0)$, $(1,0,0)$, $(0,0,1)$ and $(0,1,1)$. Although
there are four exponents, they lie on the unique plane $x+z=1$. Thus we define
the weight of $(\text{P}_{\text{V}})$ by $(1,0,1,1)$ and use the weighted
projective space $\mathbb{C}P^{3}(1,0,1,1)$.

The sixth Painlevé equation is given by

$\displaystyle(\text{P}_{\text{VI}})\colon\
\begin{cases}z(z-1)x^{\prime}=-3x^{2}y^{2}+2x^{2}y+2x^{2}yz-x^{2}z+g_{1},\\\
z(z-1)y^{\prime}=2xy^{3}-2xy^{2}-2xy^{2}z+2xyz+g_{2},\end{cases}$

where $(g_{1},g_{2})$ consists of terms including parameters. Exponents of the
other terms are given by $(1,2,0)$, $(1,1,0)$, $(1,1,1)$ and $(1,0,1)$, which
lie on the unique plane $x=1$. Thus we define the weight of
$(\text{P}_{\text{VI}})$ as $(1,0,0,1)$ and use the weighted projective space
$\mathbb{C}P^{3}(1,0,0,1)$.

| $(p,q,r,s)$ | $\deg(H)$ | $\kappa$ |  $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ | $l$ | $c$
---|---|---|---|---|---|---
$\text{P}_{\text{I}}$ | $(3,2,4,5)$ | 6 | 6 | $6,4,5$ | 1 | 2
$\text{P}_{\text{II}}$ | $(2,1,2,3)$ | 4 | 4 | $4,2,3$ | 2 | 3
$\text{P}_{\text{IV}}$ | $(1,1,1,2)$ | 3 | 3 | $3,1,2$ | 3 | 4
$\text{P}_{\text{III}(D_{8})}$ | $(-1,2,4,1)$ | 2 | 2 | $2,4,1$ | 2 | 3
$\text{P}_{\text{III}(D_{7})}$ | $(-1,2,3,1)$ | 2 | 2 | $2,3,1$ | 2 | 3
$\text{P}_{\text{III}(D_{6})}$ | $(0,1,2,1)$ | 2 | 2 | $2,2,1$ | 3 | 4
$\text{P}_{\text{V}}$ | $(1,0,1,1)$ | 2 | 2 | $2,1,1$ | 4 | 5
$\text{P}_{\text{VI}}$ | $(1,0,0,1)$ | 2 | 2 | $2,0,1$ | 5 | 6
Table 1: $\deg(H)$ denotes the weighted degree of the Hamiltonian function
with respect to the weight $\deg(x)=p$, $\deg(y)=q$, $\deg(z)=r$. $\kappa$
denotes the Kovalevskaya exponent defined in Section 2.2.
$(\lambda_{1},\lambda_{2},\lambda_{3})$ is the weight for the weighted blow-
up, which coincides with the eigenvalues of the Jacobi matrix at the
singularity. In [2], it is proved that $\deg(H)=\kappa=\lambda_{1}$ and
$r=\lambda_{2}$, $s=\lambda_{3}$. $l$ gives the number of types of Laurent
series solutions given in Section 2.2, and $c$ is the number of local charts
for the space of initial conditions. We always have $c=l+1$.

In this paper, the third, fifth and sixth Painlevé equations are studied with
the aid of the weighted projective spaces $\mathbb{C}P^{3}(p,q,r,s)$ and
dynamical systems theory. The weighted projective space is decomposed into the
disjoint sum
$\mathbb{C}P^{3}(p,q,r,s)\simeq\mathbb{C}P^{2}(p,q,r)\cup\mathbb{C}^{3}$. This
implies that the natural phase space $\mathbb{C}^{3}=\\{(x,y,z)\\}$ of the
Painlevé equations is embedded in $\mathbb{C}P^{3}(p,q,r,s)$ and the two-
dimensional manifold $\mathbb{C}P^{2}(p,q,r)$ is attached at infinity. We
regard the Painlevé equation as a three-dimensional autonomous vector field
defined on $\mathbb{C}P^{3}(p,q,r,s)$. Then, the vector field has several
fixed points on the infinity set $\mathbb{C}P^{2}(p,q,r)$. These fixed points
describe the asymptotic behavior of solutions. Some of these fixed points
correspond to movable singularities, and the other correspond to the irregular
singular point.

The dynamical systems theory is applied to these fixed points to investigate
the Painlevé equations. The singular normal form of the Painlevé equation [3,
6], which is a local integrable system around a movable singularity, is
obtained by applying the normal form theory around fixed points. The space of
initial conditions and its symplectic atlas are constructed by the weighted
blow-up at these fixed points. The weight for the weighted blow-up, which is
also an invariant of the Painlevé equation related to the Kovalevskaya
exponent [2, 3], is determined by eigenvalues of the Jacobi matrix of the
vector field at the fixed points. It is known that the Painlevé equations are
reduced to the Riccati equations when the parameters take certain specific
values. Such Riccati solutions are characterized as a center (un)stable
manifold at the fixed point on $\mathbb{C}P^{3}(p,q,r,s)$. Although some of
these results are well known for experts, our new approach based on the
weighted projective space and dynamical systems theory provides a systematic
way to investigate them. From our analysis, it turns out that the weights and
the Kovalevskaya exponents are important invariants of the Painlevé equations.
In particular, the Painlevé equations may be classified by these invariants,
which will be reported in a forthcoming paper.

Our method will be explained in detail for the third Painlevé equation of type
$D_{6}$ in Section 3. Since the strategy for the other Painlevé equations
$(\text{P}_{\text{III}(D_{7})})$, $(\text{P}_{\text{III}(D_{8})})$,
$(\text{P}_{\text{V}})$ and $(\text{P}_{\text{VI}})$ is completely the same as
that for $(\text{P}_{\text{III}(D_{6})})$, we only show a sketch and several
formulae for them after Section 4. See [3] for $(\text{P}_{\text{I}})$,
$(\text{P}_{\text{II}})$ and $(\text{P}_{\text{IV}})$.

##  2 Settings

###  2.1 Weighted projective spaces

For a tuple of integers $(p_{1},\dots,p_{m},r,s)$, consider the weighted
$\mathbb{C}^{*}$-action on $\mathbb{C}^{m+2}$ defined by

$\displaystyle(x_{1},\dots,x_{m},z,\varepsilon)\mapsto\big{(}\lambda^{p_{1}}x_{1},\dots,\lambda^{p_{m}}x_{m},\lambda^{r}z,\lambda^{s}\varepsilon\big{)},\qquad\lambda\in\mathbb{C}^{*}:=\mathbb{C}\backslash\\{0\\}.$

The quotient space

$\displaystyle\mathbb{C}P^{m+1}(p_{1},\dots,p_{m},r,s):=\mathbb{C}^{m+2}\backslash\\{0\\}/\mathbb{C}^{*}$

gives an $(m+1)$-dimensional orbifold called the weighted projective space. In
this paper, we only use a three-dimensional space. When $(p,q,r,s)$ are
positive integers, the orbifold structure of $\mathbb{C}P^{3}(p,q,r,s)$ is
obtained as follows:

The space $\mathbb{C}P^{3}(p,q,r,s)$ is defined by the equivalence relation on
$\mathbb{C}^{4}\backslash\\{0\\}$

$\displaystyle(x,y,z,\varepsilon)\sim\big{(}\lambda^{p}x,\lambda^{q}y,\lambda^{r}z,\lambda^{s}\varepsilon\big{)}.$

(i) When $x\neq 0$,

$\displaystyle(x,y,z,\varepsilon)\sim\big{(}1,x^{-q/p}y,x^{-r/p}z,x^{-s/p}\varepsilon\big{)}=:(1,Y_{1},Z_{1},\varepsilon_{1}).$

Due to the choice of the branch of $x^{1/p}$, we also obtain

$\displaystyle(Y_{1},Z_{1},\varepsilon_{1})\sim\big{(}e^{-2q\pi
i/p}Y_{1},e^{-2r\pi i/p}Z_{1},e^{-2s\pi i/p}\varepsilon_{1}\big{)},$

by putting $x\mapsto e^{2\pi i}x$. This implies that the subset of
$\mathbb{C}P^{3}(p,q,r,s)$ such that $x\neq 0$ is homeomorphic to
$\mathbb{C}^{3}/\mathbb{Z}_{p}$, where the $\mathbb{Z}_{p}$-action is defined
as above.

(ii) When $y\neq 0$,

$\displaystyle(x,y,z,\varepsilon)\sim\big{(}y^{-p/q}x,1,y^{-r/q}z,y^{-s/q}\varepsilon\big{)}=:(X_{2},1,Z_{2},\varepsilon_{2}).$

Because of the choice of the branch of $y^{1/q}$, we obtain

$\displaystyle(X_{2},Z_{2},\varepsilon_{2})\sim\big{(}e^{-2p\pi
i/q}X_{2},e^{-2r\pi i/q}Z_{2},e^{-2s\pi i/q}\varepsilon_{2}\big{)}.$

Hence, the subset of $\mathbb{C}P^{3}(p,q,r,s)$ with $y\neq 0$ is homeomorphic
to $\mathbb{C}^{3}/\mathbb{Z}_{q}$.

(iii) When $z\neq 0$,

$\displaystyle(x,y,z,\varepsilon)\sim\big{(}z^{-p/r}x,z^{-q/r}y,1,z^{-s/r}\varepsilon\big{)}=:(X_{3},Y_{3},1,\varepsilon_{3}).$

Similarly, the subset $\\{z\neq 0\\}\subset\mathbb{C}P^{3}(p,q,r,s)$ is
homeomorphic to $\mathbb{C}^{3}/\mathbb{Z}_{r}$.

(iv) When $\varepsilon\neq 0$,

$\displaystyle(x,y,z,\varepsilon)\sim\big{(}\varepsilon^{-p/s}x,\varepsilon^{-q/s}y,\varepsilon^{-r/s}z,1\big{)}=:(X_{4},Y_{4},Z_{4},1).$

The subset $\\{\varepsilon\neq 0\\}\subset\mathbb{C}P^{3}(p,q,r,s)$ is
homeomorphic to $\mathbb{C}^{3}/\mathbb{Z}_{s}$.

This proves that the orbifold structure of $\mathbb{C}P^{3}(p,q,r,s)$ is given
by

$\displaystyle\mathbb{C}P^{3}(p,q,r,s)=\mathbb{C}^{3}/\mathbb{Z}_{p}\cup\mathbb{C}^{3}/\mathbb{Z}_{q}\cup\mathbb{C}^{3}/\mathbb{Z}_{r}\cup\mathbb{C}^{3}/\mathbb{Z}_{s}.$

The local charts $(Y_{1},Z_{1},\varepsilon_{1})$,
$(X_{2},Z_{2},\varepsilon_{2})$, $(X_{3},Y_{3},\varepsilon_{3})$ and
$(X_{4},Y_{4},Z_{4})$ defined above are called inhomogeneous coordinates as
the usual projective space. Note that they give coordinates on the lift
$\mathbb{C}^{3}$, not on the quotient $\mathbb{C}^{3}/\mathbb{Z}_{i}$
$(i=p,q,r,s)$. Therefore, any equations written in these inhomogeneous
coordinates should be invariant under the corresponding $\mathbb{Z}_{i}$
actions.

In what follows, we use the notation $(x,y,z)$ for the fourth local chart
instead of $(X_{4},Y_{4},Z_{4})$ because the Painlevé equation will be given
on this chart. The transformations between inhomogeneous coordinates are give
by

$\displaystyle
x=\varepsilon_{1}^{-p/s}=X_{2}\varepsilon_{2}^{-p/s}=X_{3}\varepsilon_{3}^{-p/s},\qquad
y=Y_{1}\varepsilon_{1}^{-q/s}=\varepsilon_{2}^{-q/s}=Y_{3}\varepsilon_{3}^{-q/s},$
$\displaystyle
z=Z_{1}\varepsilon_{1}^{-r/s}=Z_{2}\varepsilon_{2}^{-r/s}=\varepsilon_{3}^{-r/s}.$
(2.1)

The same transformation rule holds even if $p$, $q$, $r$, $s$ include negative
integers. If there are $0$ among them, for example if $p=0$, then we have
$\mathbb{C}P^{3}(0,q,r,s)\simeq\mathbb{C}\times\mathbb{C}P^{2}(q,r,s)$.
$\mathbb{C}P^{3}(p,q,r,s)$ is compact if and only if all $p$, $q$, $r$, $s$
are positive.

###  2.2 Laurent series solutions and Kovalevskaya exponents

To construct the space of initial conditions, we need the expressions of the
Laurent series of solutions. Let $(p,q,r,s)$ be the weight of a given system
determined by the Newton polyhedron. Suppose that the system has a Laurent
series solution of the form

$\displaystyle x=\sum^{\infty}_{n=0}A_{n}(z-z_{0})^{-p+n},\qquad
y=\sum^{\infty}_{n=0}B_{n}(z-z_{0})^{-q+n},$

where $(A_{0},B_{0})\neq(0,0)$ and $z_{0}$ is a movable pole. Such a Laurent
series solution is called regular. A Laurent series solution is called
exceptional if it is not expressed in this form; i.e., $(A_{0},B_{0})=(0,0)$
or the order of a pole of either $x$ or $y$ is larger than $p$ or $q$,
respectively. If a regular Laurent series represents a general solution of the
system, it includes an arbitrary parameter, which depends on initial
conditions, other than $z_{0}$. The smallest integer $\kappa$ such that
$(A_{\kappa},B_{\kappa})$ includes an arbitrary parameter is called the
Kovalevskaya exponent. In [2], it is proved that the Kovalevskaya exponent of
the regular Laurent series solution is invariant under a certain class of
coordinates transformations including the automorphism group of
$\mathbb{C}P^{3}(p,q,r,s)$. For the first, second and fourth Painlevé
equations, all Laurent series solutions are regular because they satisfy the
assumptions (A1) and (A2). The third, fifth and sixth Painlevé equations have
exceptional Laurent series solutions, however, they can be converted into the
regular series by the Bäcklund transformations. Hence, the Kovalevskaya
exponents $\kappa$ of exceptional Laurent series solutions are well-defined
and given as in Table 1. In what follows, denote $T:=z-z_{0}$.

($\text{P}_{\text{III}(D_{6})}$): The third Painlevé equation of type $D_{6}$
has three types of Laurent series solutions given by

$\displaystyle\mathrm{(i)}\quad$ $\displaystyle x=1+\frac{\beta}{z_{0}}\cdot
T+A_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle y=-z_{0}\cdot
T^{-1}+\frac{1}{2}(-1-\alpha+\beta)+B_{2}\cdot T+O\big{(}T^{2}\big{)},$
$\displaystyle\mathrm{(ii)}\quad$ $\displaystyle x=0\cdot
T^{0}-\frac{\alpha}{z_{0}}\cdot T+A_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$
$\displaystyle y=z_{0}\cdot T^{-1}+\frac{1}{2}(1-\alpha+\beta)+B_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle\mathrm{(iii)}\quad$ $\displaystyle
x=-z_{0}\cdot T^{-2}+0\cdot T^{-1}+A_{2}+O(T),$ $\displaystyle
y=-T+\frac{-2+\alpha+\beta}{2z_{0}}\cdot T^{2}+B_{2}\cdot
T^{3}+O\big{(}T^{4}\big{)},$

where $A_{2}$ is an arbitrary constant and $B_{2}$ is a certain function of
$A_{2}$. Since $(p,q)=(0,1)$, the first two series are regular, while the last
one is exceptional. The Kovalevskaya exponents of all series are $\kappa=2$.

($\text{P}_{\text{III}(D_{7})}$): The third Painlevé equation of type $D_{7}$
has two types of Laurent series solutions given by

$\displaystyle\mathrm{(i)}\quad$ $\displaystyle x=\frac{1}{z_{0}}\cdot
T+\frac{\alpha-1}{2z_{0}^{2}}\cdot T^{2}+A_{2}\cdot
T^{3}+O\big{(}T^{4}\big{)},$ $\displaystyle y=-z_{0}^{2}\cdot
T^{-2}-z_{0}\cdot T^{-1}+B_{2}+O(T),$ $\displaystyle\mathrm{(ii)}$
$\displaystyle x=-z_{0}\cdot T^{-2}+0\cdot T^{-1}+A_{2}+O(T),$ $\displaystyle
y=-T^{1}+\frac{\alpha-2}{2z_{0}}\cdot T^{2}+B_{2}\cdot
T^{3}+O\big{(}T^{4}\big{)},$

where $A_{2}$ is an arbitrary constant and $B_{2}$ is a certain function of
$A_{2}$. Since $(p,q)=(-1,2)$, the former series is regular, while the latter
one is exceptional. The Kovalevskaya exponents of both series are $\kappa=2$.

($\text{P}_{\text{III}(D_{8})}$): The third Painlevé equation of type $D_{8}$
has two types of Laurent series solutions given by

$\displaystyle\mathrm{(i)}\quad$ $\displaystyle x=-\frac{1}{2z_{0}}\cdot
T+\frac{1}{4z_{0}^{2}}\cdot T^{2}+\frac{2B_{2}-1}{4z_{0}^{2}}\cdot
T^{3}+O\big{(}T^{4}\big{)},$ $\displaystyle y=2z_{0}^{2}\cdot
T^{-2}+2z_{0}\cdot T^{-1}+B_{2}+O(T),$ $\displaystyle\mathrm{(ii)}\quad$
$\displaystyle x=2z_{0}^{2}\cdot T^{-3}+2z_{0}\cdot T^{-2}+0\cdot
T^{-1}+O(1),$ $\displaystyle y=\frac{1}{2z_{0}}T^{2}+0\cdot T^{3}+B_{2}\cdot
T^{4}+O\big{(}T^{5}\big{)},$

where $B_{2}$ is an arbitrary constant. Since $(p,q)=(-1,2)$, the former
series is regular, while the latter one is exceptional. The Kovalevskaya
exponents of both series are $\kappa=2$.

($\text{P}_{\text{V}}$): The fifth Painlevé equation has four types of Laurent
series solutions given by

$\displaystyle\mathrm{(i)}\quad$ $\displaystyle x=z_{0}\cdot
T^{-1}+\frac{1}{2}(1-z_{0}+\alpha_{1}-\alpha_{3})+A_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle y=1+\frac{\alpha_{3}}{z_{0}}\cdot
T+B_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle\mathrm{(ii)}\quad$
$\displaystyle x=-z_{0}\cdot
T^{-1}+\frac{1}{2}(-1-z_{0}+\alpha_{1}-\alpha_{3})+A_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle y=0\cdot
T^{0}-\frac{\alpha_{1}}{z_{0}}\cdot T+B_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$
$\displaystyle\mathrm{(iii)}\quad$ $\displaystyle
x=-z_{0}+(\alpha_{1}+\alpha_{2}+\alpha_{3}-2)\cdot T+A_{2}\cdot
T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle
y=T^{-1}+\frac{z_{0}+\alpha_{1}+2\alpha_{2}+\alpha_{3}-2}{2z_{0}}+B_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle\mathrm{(iv)}\quad$ $\displaystyle
x=0\cdot T^{0}+\alpha_{2}\cdot T+A_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$
$\displaystyle
y=-T^{-1}+\frac{z_{0}+\alpha_{1}+2\alpha_{2}+\alpha_{3}}{2z_{0}}+B_{2}\cdot
T+O\big{(}T^{2}\big{)},$

where $A_{2}$ is an arbitrary constant and $B_{2}$ is a certain function of
$A_{2}$. Since $(p,q)=(1,0)$, (i) and (ii) are regular, while (iii) and (iv)
are exceptional. The Kovalevskaya exponents of all series are $\kappa=2$.

($\text{P}_{\text{VI}}$): The sixth Painlevé equation has five types of
Laurent series solutions given by

$\displaystyle\mathrm{(i)}\quad$ $\displaystyle
x=T^{-1}+\frac{2-4z_{0}+\alpha_{0}-2z_{0}\alpha_{0}+z_{0}\alpha_{3}-\alpha_{4}+z_{0}\alpha_{4}}{2z_{0}(z_{0}-1)}+A_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle y=z_{0}+(2+\alpha_{0})\cdot
T+B_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle\mathrm{(ii)}\quad$
$\displaystyle x=-z_{0}\cdot
T^{-1}+\frac{1-z_{0}-\alpha_{0}+2\alpha_{3}-z_{0}\alpha_{3}-\alpha_{4}+z_{0}\alpha_{4}}{2(z_{0}-1)}+A_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle y=1-\frac{\alpha_{3}}{z_{0}}\cdot
T+B_{2}\cdot T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle\mathrm{(iii)}\quad$
$\displaystyle x=(z_{0}-1)\cdot
T^{-1}+\frac{z_{0}-\alpha_{0}-z_{0}\alpha_{3}+\alpha_{4}+z_{0}\alpha_{4}}{2z_{0}}+A_{2}\cdot
T+O\big{(}T^{2}\big{)},$ $\displaystyle y=0\cdot
T^{0}+\frac{\alpha_{4}}{z_{0}-1}\cdot T+B_{2}\cdot
T^{2}+O\big{(}T^{3}\big{)},$ $\displaystyle\mathrm{(iv)}\quad$ $\displaystyle
x=-\frac{\alpha_{1}(\alpha_{1}+\alpha_{2})}{z_{0}(z_{0}-1)}\cdot
T+O\big{(}T^{2}\big{)},\qquad y=\frac{z_{0}(z_{0}-1)}{\alpha_{1}}\cdot
T^{-1}+O(1),$ $\displaystyle\mathrm{(v)}\quad$ $\displaystyle
x=\frac{\alpha_{1}\alpha_{2}}{z_{0}(z_{0}-1)}\cdot
T+O\big{(}T^{2}\big{)},\qquad y=-\frac{z_{0}(z_{0}-1)}{\alpha_{1}}\cdot
T^{-1}+O(1),$

where $A_{2}$ is an arbitrary constant and $B_{2}$ is a certain function of
$A_{2}$. Since $(p,q)=(1,0)$, (i), (ii) and (iii) are regular, while (iv) and
(v) are exceptional. The Kovalevskaya exponents of all series are $\kappa=2$.

For all Painlevé equations, the number of types of Laurent series solutions is
smaller than the number of local charts of the space of initial conditions by
one, see Table 1.

##  3 The third Painlevé equation of type $\boldsymbol{D_{6}}$

###  3.1 $\boldsymbol{\text{P}_{\text{III}(D_{6})}}$ on
$\boldsymbol{\mathbb{C}P^{3}(0,1,2,1)}$

The orbifold structure of $\mathbb{C}P^{3}(0,1,2,1)$ is given by

$\displaystyle\mathbb{C}P^{3}(0,1,2,1)=\mathbb{C}\times\mathbb{C}P^{2}(1,2,1)=\mathbb{C}\times\big{(}\mathbb{C}^{2}\cup\mathbb{C}^{2}/\mathbb{Z}_{2}\cup\mathbb{C}^{2}\big{)}=\mathbb{C}^{3}\cup\big{(}\mathbb{C}\times\mathbb{C}^{2}/\mathbb{Z}_{2}\big{)}\cup\mathbb{C}^{3}.$

Thus, the space is covered by three inhomogeneous coordinates
$(X_{2},Z_{2},\varepsilon_{2}),(X_{3},Y_{3},\varepsilon_{3})$ and $(x,y,z)$
related as

$\displaystyle x=X_{2}=X_{3},\qquad
y=\varepsilon_{2}^{-1}=Y_{3}\varepsilon_{3}^{-1},\qquad
z=Z_{2}\varepsilon_{2}^{-2}=\varepsilon_{3}^{-2}$ (3.1)

(the first local chart $(Y_{1},Z_{1},\varepsilon_{1})$ does not appear because
$p=0$).

We give the third Painlevé equation of type $D_{6}$ on the local chart
$(x,y,z)$. On the other local charts, $(\text{P}_{\text{III}(D_{6})})$ is
expressed as

$\displaystyle\frac{dX_{2}}{d\varepsilon_{2}}=\frac{2X_{2}-2X_{2}^{2}+\varepsilon_{2}(\alpha-\alpha
X_{2}-\beta
X_{2})}{\varepsilon_{2}(1-2X_{2}-Z_{2}-\alpha\varepsilon_{2}-\beta\varepsilon_{2})},$
$\displaystyle\frac{dZ_{2}}{d\varepsilon_{2}}=\frac{2Z_{2}-4X_{2}Z_{2}-2Z_{2}^{2}+\varepsilon_{2}Z_{2}(1-2\alpha-2\beta)}{\varepsilon_{2}(1-2X_{2}-Z_{2}-\alpha\varepsilon_{2}-\beta\varepsilon_{2})},$
$\displaystyle\frac{dX_{3}}{d\varepsilon_{3}}=-\frac{2}{\varepsilon_{3}^{2}}\left(2X_{3}Y_{3}-2X_{3}^{2}Y_{3}+\varepsilon_{3}(\alpha-\alpha
X_{3}-\beta X_{3})\right),$
$\displaystyle\frac{dY_{3}}{d\varepsilon_{3}}=-\frac{2}{\varepsilon_{3}^{2}}\left(1-Y_{3}^{2}+2X_{3}Y_{3}^{2}+\varepsilon_{3}Y_{3}\left(-\frac{1}{2}+\alpha+\beta\right)\right).$

In order to apply dynamical systems theory later, it is convenient to rewrite
them as $3$-dimensional autonomous vector fields of the form

$\displaystyle\dot{X}_{2}=2X_{2}-2X_{2}^{2}+\varepsilon_{2}(\alpha-\alpha
X_{2}-\beta X_{2}),$
$\displaystyle\dot{Z}_{2}=2Z_{2}-4X_{2}Z_{2}-2Z_{2}^{2}+\varepsilon_{2}Z_{2}(1-2\alpha-2\beta),$
$\displaystyle\dot{\varepsilon}_{2}=\varepsilon_{2}(1-2X_{2}-Z_{2}-\alpha\varepsilon_{2}-\beta\varepsilon_{2}),$
(3.2)

and

$\displaystyle\dot{X}_{3}=2X_{3}Y_{3}-2X_{3}^{2}Y_{3}+\varepsilon_{3}(\alpha-\alpha
X_{3}-\beta X_{3}),$
$\displaystyle\dot{Y}_{3}=1-Y_{3}^{2}+2X_{3}Y_{3}^{2}+\varepsilon_{3}Y_{3}(-\frac{1}{2}+\alpha+\beta),$
$\displaystyle\dot{\varepsilon}_{3}=-\frac{\varepsilon_{3}^{2}}{2},$ (3.3)

where $(\dot{\,\,})=d/dt$ and $t\in\mathbb{C}$ parameterizes each integral
curve.

We also have the decomposition

$\displaystyle\mathbb{C}P^{3}(0,1,2,1)=\mathbb{C}P^{2}(0,1,2)\cup\mathbb{C}^{3}\qquad(\text{disjoint}).$

Since $\mathbb{C}P^{2}(0,1,2)=\mathbb{C}\times\mathbb{C}P^{1}(1,2)$ and
$\mathbb{C}P^{1}(1,2)$ is the usual projective line,

$\displaystyle\mathbb{C}P^{3}(0,1,2,1)=\big{(}\mathbb{C}\times\mathbb{C}P^{1}\big{)}\cup\mathbb{C}^{3},$

where $\mathbb{C}^{3}=\\{(x,y,z)\\}$ and
$\mathbb{C}\times\mathbb{C}P^{1}=\\{(X_{2},Z_{2},0)\\}\cup\\{(X_{3},Y_{3},0)\\}$
in coordinates. This implies that the set $\mathbb{C}\times\mathbb{C}P^{1}$ is
attached at infinity of the natural phase space $\mathbb{C}^{3}=\\{(x,y,z)\\}$
of $(\text{P}_{\text{III}(D_{6})})$, and the asymptotic behavior of solutions
can be studied by the limit $\varepsilon_{2}\to 0$ or $\varepsilon_{3}\to 0$.

The vector fields (3.2) and (3.3) have exactly four fixed points on the
infinity set $\\{\varepsilon_{2}=0\\}\cup\\{\varepsilon_{3}=0\\}$ given by

$\displaystyle(X_{2},Z_{2},\varepsilon_{2})=(0,0,0),\ (1,0,0),\ (0,1,0),\
(1,-1,0).$

The Jacobi matrices of (3.2) at the fixed points are

$\displaystyle\left(\begin{matrix}2&0&\alpha\\\ 0&2&0\\\
0&0&1\end{matrix}\right),\quad-\left(\begin{matrix}2&0&\beta\\\ 0&2&0\\\
0&0&1\end{matrix}\right),\quad\left(\begin{matrix}2&0&\alpha\\\
-4&-2&\gamma\\\
0&0&0\end{matrix}\right),\quad-\left(\begin{matrix}2&0&\beta\\\
-4&-2&\gamma\\\ 0&0&0\end{matrix}\right),$ (3.4)

respectively, where $\gamma=1-2\alpha-2\beta$. The latter two points, for
which $Z_{2}\neq 0\Rightarrow z=\infty$, correspond to the irregular singular
point. Since the Jacobi matrix has a zero eigenvalue, there exists a one-
dimensional center manifold at each fixed point. The asymptotic expansion of
the center manifold [5] yields the asymptotic expansion of $(x(z),y(z))$ as
$|z|\to\infty$.

On the other hand, the former two fixed points correspond to movable poles.
Indeed, it is easy to verify by using (3.1) that the regular Laurent series
solutions (ii) and (i) given in Section 2.2 converge to the points
$(X_{2},Z_{2},\varepsilon_{2})=(0,0,0)$ and $(1,0,0)$, respectively, as $z\to
z_{0}$. The exceptional Laurent series solution (iii) does not converge to
some point on $\mathbb{C}P^{3}(0,1,2,1)$ as $z\to z_{0}$; the space
$\mathbb{C}P^{3}(0,1,2,1)$ is not compact.

To treat the Laurent series (iii), we use the Bäcklund transformations. See
[9] for the complete list of the Bäcklund transformations of two-dimensional
Painlevé equations. It is known that the transformation groups of the Painlevé
equations are isomorphic to the extended affine Weyl groups. For a classical
root system $R$, the affine Weyl group and the extended affine Weyl group are
denoted by $W(R^{(1)})$ and $\widetilde{W}(R^{(1)})$, respectively. Let
$G=\operatorname{Aut}(R^{(1)})$ be the Dynkin automorphism group of the
extended Dynkin diagram. We have $\widetilde{W}(R^{(1)})\cong G\ltimes
W(R^{(1)})$. For the third Painlevé equation of type $D_{6}$, $R=2A_{1}$ and

$\displaystyle\widetilde{W}\big{(}(2A_{1})^{(1)}\big{)}\cong G\ltimes
W\big{(}(2A_{1})^{(1)}\big{)},\qquad W\big{(}(2A_{1})^{(1)}\big{)}=\langle
s_{0},s_{1},s_{0}^{\prime},s_{1}^{\prime}\rangle,$ $\displaystyle
G=\operatorname{Aut}\big{(}D_{6}^{(1)}\big{)}=\operatorname{Aut}\big{(}(2A_{1})^{(1)}\big{)}=\langle\pi_{1},\pi_{2},\sigma_{1}\rangle.$

The action of each element is given in Table 2, where
$f_{0}=x+(\alpha+\beta-1)/y+z/y^{2}$, $f_{1}=\beta y+(x-1)y^{2}+z$,
$f_{2}=\alpha y+xy^{2}+z$ and $f_{3}=2xy+\alpha+\beta-1$.

| $\alpha$ | $\beta$ | $y$ | $x$ | $z$
---|---|---|---|---|---
$s_{0}$ |  $2-\alpha$  | $\beta$ | $\displaystyle y+\frac{1-\alpha}{f_{0}-1}$ | $\displaystyle x-\frac{(1-\alpha)(f_{3}-2y)}{f_{1}}-\frac{(1-\alpha)^{2}z}{f_{1}^{2}}$ | $z$
$s_{1}$ | $-\alpha$ | $\beta$ | $\displaystyle y+\frac{\alpha}{x}$ | $x$ | $z$
$s_{0}^{\prime}$ | $\alpha$ | $2-\beta$ | $\displaystyle y+\frac{1-\beta}{f_{0}}$ | $\displaystyle x-\frac{(1-\beta)f_{3}}{f_{2}}-\frac{(1-\beta)^{2}z}{f_{2}^{2}}$ | $z$
$s_{1}^{\prime}$ | $\alpha$ | $-\beta$ | $\displaystyle y+\frac{\beta}{x-1}$ | $x$ | $z$
$\pi_{1}$ | $1-\alpha$ | $\beta$ | $\displaystyle-\frac{z}{y}$ | $\displaystyle\frac{y}{z}(xy-y+\beta)+1$ | $z$
$\pi_{2}$ | $\alpha$ | $1-\beta$ | $\displaystyle\frac{z}{y}$ | $\displaystyle-\frac{y}{z}(xy+\alpha)$ | $z$
$\sigma_{1}$ | $\beta$ | $\alpha$ | $-y$ | $1-x$ | $-z$
Table 2: The action of the extended affine Weyl group for
$(\text{P}_{\text{III}(D_{6})})$.

Let us consider another space $\mathbb{C}P^{3}(0,1,2,1)$ with inhomogeneous
coordinates
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})$,
$(\widetilde{X}_{3},\widetilde{Y}_{3},\widetilde{\varepsilon}_{3})$ and
$(\widetilde{x},\widetilde{y},\widetilde{z})$ satisfying the same formula as
(3.1). We glue two copies of $\mathbb{C}P^{3}(0,1,2,1)$ by the transformation
$\pi_{2}$

$\displaystyle(\widetilde{x},\widetilde{y},\widetilde{z},\widetilde{\alpha},\widetilde{\beta})=\pi_{2}(x,y,z,\alpha,\beta)=\left(-\frac{y}{z}(xy+\alpha),\frac{z}{y},z,\alpha,1-\beta\right),$
(3.5)

which defines a manifold denoted by $\mathcal{M}_{1}$. Since
$(\text{P}_{\text{III}(D_{6})})$ is invariant under the action of $\pi_{2}$,
the system written in the
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})$-chart also
satisfies equation (3.2), in which $(\alpha,\beta)$ is replaced by
$(\widetilde{\alpha},\widetilde{\beta})=(\alpha,1-\beta)$. By the formula
(3.5) or

$\displaystyle\widetilde{X}_{2}=-\big{(}xy^{2}+\alpha
y\big{)}/z,\qquad\widetilde{Z}_{2}=y^{2}/z,\qquad\widetilde{\varepsilon}_{2}=y/z,$

it is easy to show that the exceptional Laurent series (iii) is converted to
the regular series (i) in the
$(\widetilde{x},\widetilde{y},\widetilde{z})$-chart, and it approaches to the
fixed point
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})=(1,0,0)$ as
$z\to z_{0}$.

We have proved that Laurent series solutions (i), (ii) and (iii) of
$(\text{P}_{\text{III}(D_{6})})$ converge to the fixed points
$(X_{2},Z_{2},\varepsilon_{2})=(1,0,0),(0,0,0)$ and
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})=(1,0,0)$ on
$\mathcal{M}_{1}$, respectively, as $z\to z_{0}$. Hence, the study of movable
poles are reduced to the study of the fixed points and dynamical systems
theory is applicable to investigate them.

As a simple application, we can prove the next theorem.

######  Theorem 3.1.

There exists a local analytic transformation
$(X_{2},Z_{2},\varepsilon_{2})\mapsto(u,v,w)$ defined near
$(X_{2},Z_{2},\varepsilon_{2})=(0,0,0)$ such that equation (3.2) is
transformed into the linearized system

$\displaystyle\dot{u}=2u+\alpha w,\qquad\dot{v}=2v,\qquad\dot{w}=w.$

Similarly, the vector field is locally linearized around
$(X_{2},Z_{2},\varepsilon_{2})=(1,0,0)$ and
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})=(1,0,0)$.

This result implies that $(\text{P}_{\text{III}(D_{6})})$ is locally
transformed to the integrable system around each movable singularity. A proof
is a straightforward application of Poincaré’s linearization theorem [5] in
dynamical systems theory. See [3] for the detail, in which a similar result is
proved for the first, second and fourth Painlevé equations, and also [2], in
which it is proved that any differential equations having the Painlevé
property is locally linearizable. Such a local integrable system for the
Painlevé equation is called the singular normal form in [3, 6].

###  3.2 The space of initial conditions

A purpose in this section is to construct the space of initial conditions for
$(\text{P}_{\text{III}(D_{6})})$. On the manifold $\mathcal{M}_{1}$, there are
three singularities of the foliation of integral curves;
$(X_{2},Z_{2},\varepsilon_{2})=(1,0,0),(0,0,0)$ and
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})=(1,0,0)$.
They correspond to movable poles of the Laurent series solutions (i), (ii) and
(iii), respectively. We will resolve these singularities by weighted blow-ups.
On the blow-up space, $(\text{P}_{\text{III}(D_{6})})$ is again written in a
Hamiltonian system, whose Hamiltonian function is polynomial in dependent
variables. This implies that singularities of the foliation are resolved and
the space of initial conditions is obtained by three times blow-ups of
$\mathcal{M}_{1}$. This strategy is applicable to the other Painlevé
equations, even for higher-dimensional Painlevé equations.

(i) blow-up at $(X_{2},Z_{2},\varepsilon_{2})=(0,0,0)$.

For the vector field (3.2), put

$\displaystyle u=X_{2}+\alpha\varepsilon_{2},\qquad v=Z_{2},\qquad
w=\varepsilon_{2}.$

Then, the linear part is diagonalized and we obtain

$\displaystyle\dot{u}=2u-2u^{2}+w(\alpha u-\alpha v-\beta u),$
$\displaystyle\dot{v}=2v+v(w-4u-2v+2\alpha w-2\beta w),$
$\displaystyle\dot{w}=w-w(2u+v-\alpha w+\beta w).$

The origin $(u,v,w)=(0,0,0)$ is a singularity of the foliation of integral
curves. To resolve it, we introduce the weighted blow-up defined by

$\displaystyle u=u_{1}^{2}=v_{2}^{2}u_{2}=w_{3}^{2}u_{3},\qquad
v=u_{1}^{2}v_{1}=v_{2}^{2}=w_{3}^{2}v_{3},\qquad
w=u_{1}w_{1}=v_{2}w_{2}=w_{3}.$

The weight $(2,2,1)$, the exponents in the right hand sides, is taken from the
eigenvalues of the Jacobi matrix at the singularity. The exceptional divisor
$\\{u_{1}=0\\}\cup\\{v_{2}=0\\}\cup\\{w_{3}=0\\}$ is the weighted projective
space $\mathbb{C}P^{2}(2,2,1)$. The relation between the original coordinates
$(x,y,z)$ and the new coordinates $(u_{3},v_{3},w_{3})$ is given by

$\displaystyle x=u_{3}w_{3}^{2}-\alpha w_{3},\qquad$ $\displaystyle
y=1/w_{3},\qquad$ $\displaystyle z=v_{3},$ $\displaystyle u_{3}=xy^{2}+\alpha
y,\qquad$ $\displaystyle w_{3}=1/y,\qquad$ $\displaystyle v_{3}=z.$ (3.6)

Note that eventually the independent variable $z$ is not changed and (3.6)
defines a fiber bundle over $z$-space, whose fiber is a $(x,y)$-space and
$(u_{3},w_{3})$-space glued by the above relation. In the new coordinates,
$(\text{P}_{\text{III}(D_{6})})$ is transformed into the Hamiltonian system

$\displaystyle\frac{dw_{3}}{dz}=-\frac{\partial H_{1}}{\partial
u_{3}},\qquad\frac{du_{3}}{dz}=\frac{\partial H_{1}}{\partial w_{3}},$

with the Hamiltonian function

$\displaystyle zH_{1}=-u+u^{2}w^{2}+uw^{2}z-\alpha wz-(\alpha-\beta)uw$

(here, the subscript is omitted for simplicity). Since $H_{1}$ is polynomial
in $(u_{3},w_{3})$, there are no singularities of the foliation in this chart;
the singularity associated with the Laurent series solution (ii) is resolved.
Furthermore, we can verify that

$\displaystyle dx\wedge dy=dw_{3}\wedge du_{3},\qquad dx\wedge dy+dH\wedge
dz=dw_{3}\wedge du_{3}+dH_{1}\wedge dz.$ (3.7)

(ii) blow-up at $(X_{2},Z_{2},\varepsilon_{2})=(1,0,0)$.

Putting $X_{2}-1=\hat{X}_{2}$ and $u=\hat{X}_{2}+\beta\varepsilon_{2}$,
$v=Z_{2}$, $w=\varepsilon_{2}$ for the vector field (3.2) results in

$\displaystyle\dot{u}=-2u-2u^{2}-w(\alpha u-\beta u+\beta v),$
$\displaystyle\dot{v}=-2v+v(w-4u-2v-2\alpha w+2\beta w),$
$\displaystyle\dot{w}=-w-w(2u+v+\alpha w-\beta w).$

To resolve the singularity at $(u,v,w)=(0,0,0)$, we introduce the weighted
blow-up with the weight $(2,2,1)$

$\displaystyle u=u_{4}^{2}=v_{5}^{2}u_{5}=w_{6}^{2}u_{6},\qquad
v=u_{4}^{2}v_{4}=v_{5}^{2}=w_{6}^{2}v_{6},\qquad
w=u_{4}w_{4}=v_{5}w_{5}=w_{6}.$

The relation between the original coordinates $(x,y,z)$ and the new
coordinates $(u_{6},v_{6},w_{6})$ is given by

$\displaystyle x=u_{6}w_{6}^{2}-\beta w_{6}+1,\qquad$ $\displaystyle
y=1/w_{6},\qquad$ $\displaystyle z=v_{6},$ $\displaystyle u_{6}=xy^{2}+\beta
y-y^{2},\qquad$ $\displaystyle w_{6}=1/y,\qquad$ $\displaystyle v_{6}=z,$
(3.8)

which defines a fiber bundle over $z$-space. In the new coordinates,
$(\text{P}_{\text{III}(D_{6})})$ is written as the Hamiltonian system

$\displaystyle\frac{dw_{6}}{dz}=-\frac{\partial H_{2}}{\partial
u_{6}},\qquad\frac{du_{6}}{dz}=\frac{\partial H_{2}}{\partial w_{6}},$

with the Hamiltonian function

$\displaystyle zH_{2}=u+u^{2}w^{2}+uw^{2}z-\beta wz+(\alpha-\beta)uw$ (3.9)

(here, the subscript is omitted for simplicity). Since $H_{2}$ is polynomial
in $(u_{6},w_{6})$, the singularity associated with the Laurent series
solution (i) is resolved. As before, we can verify the symplectic relation
(3.7).

(iii) blow-up at
$(\widetilde{X}_{2},\widetilde{Z}_{2},\widetilde{\varepsilon}_{2})=(1,0,0)$.

This singularity is resolved by the same way as above by the weighted blow-up
with the weight $(2,2,1)$. The final result is easily obtained by the Bäcklund
transformation (3.5) as follows. Define the new coordinates
$(u_{9},v_{9},w_{9})$ by

$\displaystyle
u_{9}=\widetilde{x}\widetilde{y}^{2}+\widetilde{\beta}\widetilde{y}-\widetilde{y}^{2},\qquad
w_{9}=1/\widetilde{y},\qquad v_{9}=\widetilde{z},$

which is the same formula as (3.8). Substituting (3.5) yields

$\displaystyle u_{9}=-z^{2}/y^{2}-xz-(\alpha+\beta-1)z/y,\qquad
w_{9}=y/z,\qquad v_{9}=z.$ (3.10)

By this coordinates change, $(\text{P}_{\text{III}(D_{6})})$ is transformed
into the Hamiltonian system of $(u_{9},w_{9})$, whose Hamiltonian function has
the same form as (3.9), though $(\alpha,\beta)$ is replaced by
$(\widetilde{\alpha},\widetilde{\beta})=(\alpha,1-\beta)$.

In this manner, all singularities are resolved and we have

######  Theorem 3.2.

The space of initial conditions $E(z)$ of $({\rm P}_{{\rm III}(D_{6})})$ is
given by
$\mathbb{C}^{2}_{(x,y)}\cup\mathbb{C}^{2}_{(u_{3},w_{3})}\cup\mathbb{C}^{2}_{(u_{6},w_{6})}\cup\mathbb{C}^{2}_{(u_{9},w_{9})}$
glued by the symplectic transformations (3.6), (3.8) and (3.10). The space
$E(z)$ is a nonsingular symplectic surface parameterized by
$z\in\mathbb{C}\backslash\\{0\\}$, on which $({\rm P}_{{\rm III}(D_{6})})$ is
expressed as a polynomial Hamiltonian system.

###  3.3 The Riccati solutions

It is known that when the parameter $\alpha$ is an integer or $\beta$ is an
integer, there exists a one-parameter family of solutions of
$(\text{P}_{\text{III}(D_{6})})$ satisfying the Riccati equation, which is
equivalent to the Bessel equation. For example, when $\alpha=0$ (resp.
$\beta=0$), the Riccati equation is given by

$\displaystyle z\frac{dy}{dz}=-y^{2}+\beta y+z,\qquad\text{resp.}\quad
z\frac{dy}{dz}=y^{2}+\alpha y+z.$ (3.11)

The Bäcklund transformations yield the Riccati equations for a general case
$\alpha\in\mathbb{Z}$ or $\beta\in\mathbb{Z}$. Let us prove this fact from a
view point of dynamical systems theory.

Recall that there are two fixed points
$(X_{2},Z_{2},\varepsilon_{2})=(0,1,0),(1,-1,0)$ of the vector field (3.2)
corresponding to the irregular singular point. The Jacobi matrices at these
points are shown in (3.4), and eigenvalues of them are $2$, $-2$, $0$. Hence,
there exist a center-stable manifold and a center-unstable manifold at these
points. Let us calculate them explicitly.

For the point $(X_{2},Z_{2},\varepsilon_{2})=(0,1,0)$, put
$\hat{Z}_{2}=Z_{2}-1$ and

$\displaystyle
u=X_{2}+\hat{Z}_{2}+\frac{1}{2}\varepsilon_{2}(\alpha+2\beta-1),\qquad
v=-X_{2}-\frac{1}{2}\alpha\varepsilon_{2},\qquad w=\varepsilon_{2}.$

Then, the linear part of equation (3.2) is diagonalized around $(0,1,0)$ and
we obtain

$\displaystyle\dot{u}=-2u+u\left(-2u-\frac{1}{2}w-\frac{\alpha}{2}w+\beta
w\right)+(1-\alpha)\left(\frac{1}{2}vw+\frac{1}{4}w^{2}-\frac{\beta}{2}w^{2}\right),$
$\displaystyle\dot{v}=2v+v\left(2v+\frac{\alpha}{2}w-\beta
w\right)+\frac{1}{2}\alpha\left(uw+\frac{1}{2}w^{2}-\beta w^{2}\right),$
$\displaystyle\dot{w}=w\left(v-u-\frac{1}{2}w\right).$

The stable, unstable, center subspaces are $u,v,w$-directions, respectively.
In particular, the center-unstable manifold (resp. the center-stable manifold)
is expressed as the graph of some function $v=\phi(u,w)$ (resp.
$u=\varphi(v,w)$).

It is obvious that when $\alpha=0$, the center-unstable manifold is exactly
given by $v=0$. This implies $X_{2}=x=0$. When $\alpha=0$ and $x=0$, equation
(1.4) is reduced to the Riccati equation (3.11). Similarly, when $\alpha=1$,
the center-stable manifold is exactly given by $u=0$. This implies

$\displaystyle 0=X_{2}+\hat{Z}_{2}+\beta\varepsilon_{2}=x+z/y^{2}+\beta/y-1.$

Substituting this relation to equation (1.4) yields the Riccati equation for
$\alpha=1$, though it is also obtained by the Bäcklund transformation to that
for $\alpha=0$. The same argument at the point
$(X_{2},Z_{2},\varepsilon_{2})=(1,-1,0)$ provides the Riccati equation for
$\beta=0$.

######  Theorem 3.3.

When $\alpha\in\mathbb{Z}$ or $\beta\in\mathbb{Z}$, there exists a one-
parameter family of solutions of $({\rm P}_{{\rm III}(D_{6})})$ governed by
the Riccati equation of Bessel type. The one-parameter family of solutions
forms a center-$($un$)$stable manifold of (3.2) in $\mathbb{C}P^{3}(0,1,2,1)$.

The same argument is applicable to the other Painlevé equations, even for
higher-dimensional Painlevé equations. For two-dimensional equations, it is
well known that when parameters take certain specific values, the Painlevé
equations are reduced to Riccati-type equations except for
$(\text{P}_{\text{I}})$, $(\text{P}_{\text{III}(D_{7})})$ and
$(\text{P}_{\text{III}(D_{8})})$. For such specific values of parameters,
center-(un)stable manifolds in $\mathbb{C}P^{3}(p,q,r,s)$ are exactly
calculated and the Riccati equations are obtained by restricting the equations
to the center-(un)stable manifolds.

The result is summarized in Table 3. The third column shows one of the
parameters for which equations are reduced to the Riccati equations. The other
possible parameters are obtained by the Bäcklund transformations. The fourth
column denotes the name of the Riccati equation when it is written as a second
order linear equation. Note that the weight of the Riccati equation is also
defined through the Newton polyhedron. For example, the Riccati equation of
Airy type is defined by

$\displaystyle\frac{dy}{dz}=y^{2}+z.$

The exponents of monomials in the right hand side are $(1,1)$ and $(-1,2)$.
Since they are on the line $y+2z=3$, the weighted projective space for the
equation is $\mathbb{C}P^{2}(1,2,3)$. See [4] for the analysis of the Airy
equation by means of the weighted projective space. The last column of Table 3
gives the weight of each Riccati equation. It is interesting to note that
these weights are obtained by deleting the first or second numbers from the
weights of the corresponding Painlevé equations.

| weight | parameter | Riccati | weight
---|---|---|---|---
$\text{P}_{\text{II}}$ | $(2,1,2,3)$ | $\alpha=1/2$ | Airy | $(1,2,3)$
$\text{P}_{\text{IV}}$ | $(1,1,1,2)$ | $\alpha=0$ | Hermite | $(1,1,2)$
$\text{P}_{\text{III}(D_{6})}$ | $(0,1,2,1)$ | $\alpha=0$ | Bessel | $(1,2,1)$
$\text{P}_{\text{V}}$ | $(1,0,1,1)$ | $\alpha_{1}=0$ | CHG | $(1,1,1)$
$\text{P}_{\text{VI}}$ | $(1,0,0,1)$ | $\alpha_{1}=0$ | HG | $(1,0,1)$
Table 3: The type of Riccati equations and their weights. CHG and HG denote
the confluent hypergeometric and hypergeometric equations, respectively.

###  3.4 Boutroux’s coordinates

For the first and second Painlevé equations, the third local chart
$(X_{3},Y_{3},\varepsilon_{3})$ of $\mathbb{C}P^{3}(p,q,r,s)$ defined by
equation (2.1) is equivalent to Boutroux’s coordinates introduced in [1] to
investigate the irregular singular point $z=\infty$. For the other Painlevé
equations, the chart $(X_{3},Y_{3},\varepsilon_{3})$ plays the same role as
Boutroux’s coordinates; as $\varepsilon_{3}\to 0$, equation (3.3) is reduced
to the autonomous Hamiltonian system

$\displaystyle\dot{X}_{3}=2X_{3}Y_{3}-2X_{3}^{2}Y_{3},\qquad\dot{Y}_{3}=1-Y_{3}^{2}+2X_{3}Y_{3}^{2},$
(3.12)

which is often called the autonomous limit. Since a generic integral curve
given by $\mathcal{H}_{{\rm
III}(D_{6})}=X_{3}^{2}Y_{3}^{2}-X_{3}Y_{3}^{2}+X_{3}=\text{const}$ is an
elliptic curve, a general solution can be expressed by Weierstrass’s elliptic
functions. Then, the system (3.3) with small $\varepsilon_{3}$ can be studied
by a perturbation method.

Let us calculate the action of the extended affine Weyl group
$\widetilde{W}\big{(}(2A_{1})^{(1)}\big{)}$ restricted on the set
$\\{\varepsilon_{3}=0\\}$, which leaves the autonomous limit (3.12) invariant.

######  Proposition 3.4.

The birational transformation group
$\widetilde{W}\big{(}(2A_{1})^{(1)}\big{)}$ is extended to the birational
transformation group acting on $\mathbb{C}P^{3}(0,1,2,1)$. The transformation
group which leaves the autonomous limit (3.12) invariant is given by
$\mathbb{Z}_{2}\ltimes\operatorname{Aut}\big{(}(2A_{1})^{(1)}\big{)}=\mathbb{Z}_{2}\ltimes\langle\pi_{1},\pi_{2},\sigma_{1}\rangle$.

###### Proof.

The first part of Proposition is verified by a straightforward calculation. To
show the second part, we should write down the actions in the
$(X_{3},Y_{3},\varepsilon_{3})$-chart. For example, the action of $s_{1}$ in
the $(X_{3},Y_{3},\varepsilon_{3})$-chart is given by

$\displaystyle(X_{3},Y_{3},\varepsilon_{3},\alpha,\beta)\mapsto\left(X_{3},Y_{3}+\alpha\frac{\varepsilon_{3}}{X_{3}},\varepsilon_{3},-\alpha,\beta\right).$

On the set $\\{\varepsilon_{3}=0\\}$, it is reduced to

$\displaystyle(X_{3},Y_{3},\alpha,\beta)\mapsto(X_{3},Y_{3},-\alpha,\beta).$

Since the autonomous limit (3.12) is independent of the parameters $\alpha$,
$\beta$, the action of $s_{1}$ to (3.12) is trivial. Similarly, the actions of
$s_{0},s_{0}^{\prime}$ and $s_{1}^{\prime}$ to (3.12) are reduced to the
trivial one. On the other hand, it is easy to confirm that the restriction of
the actions of $\pi_{1}$, $\pi_{2}$, $\sigma_{1}$ are not trivial, which are
explicitly given by

$\displaystyle\pi_{1}\colon\
(X_{3},Y_{3})\mapsto\big{(}X_{3}Y_{3}^{2}-Y_{3}^{2}+1,-1/Y_{3}\big{)},$
$\displaystyle\pi_{2}\colon\
(X_{3},Y_{3})\mapsto\big{(}{-}X_{3}Y_{3}^{2},1/Y_{3}\big{)},$
$\displaystyle\sigma_{1}\colon\
(X_{3},Y_{3})\mapsto\big{(}1-X_{3},\sqrt{-1}Y_{3}\big{)}.$

Furthermore, (3.12) is invariant under the $\mathbb{Z}_{2}$ action
$Y_{3}\mapsto-Y_{3}$ due to the orbifold structure of
$\mathbb{C}P^{3}(0,1,2,1)$. Since $r=\deg(z)=2$,
$(X_{3},Y_{3},\varepsilon_{3})$ are coordinates on the lift $\mathbb{C}^{3}$
of the quotient $\mathbb{C}^{3}/\mathbb{Z}_{2}$, where $\mathbb{Z}_{2}$ action
is defined by
$(X_{3},Y_{3},\varepsilon_{3})\mapsto(X_{3},-Y_{3},-\varepsilon_{3})$. This is
reduced to the action $Y_{3}\mapsto-Y_{3}$ on the set
$\\{\varepsilon_{3}=0\\}$. ∎

A similar result also holds for the other Painlevé equations except for
($\text{P}_{\text{VI}}$) and summarized in Table 4. If $r=\deg(z)=1$, the
symmetry group of the autonomous limit is given by the Dynkin automorphism
group $G=\operatorname{Aut}\big{(}R^{(1)}\big{)}$ because the action of
$W\big{(}R^{(1)}\big{)}$ on the set $\\{\varepsilon_{3}=0\\}$ is reduced to
the trivial action as above. If $r>1$, the autonomous limit is further
invariant under the $\mathbb{Z}_{r}$ action arising from the orbifold
structure of $\mathbb{C}P^{3}(p,q,r,s)$. The autonomous limit on the set
$\\{\varepsilon_{3}=0\\}$ is not defined for ($\text{P}_{\text{VI}}$) in this
way because $r=0$.

| weight | $\mathcal{H}_{J}$ | symmetry
---|---|---|---
$\text{P}_{\text{I}}$ | $(3,2,4,5)$ | $X^{2}-4Y^{3}-2Y$ | $\mathbb{Z}_{4}$
$\text{P}_{\text{II}}$ | $(2,1,2,3)$ | $X^{2}-Y^{4}-Y^{2}$ | $\mathbb{Z}_{2}\ltimes\operatorname{Aut}\big{(}A_{1}^{(1)}\big{)}$
$\text{P}_{\text{IV}}$ | $(1,1,1,2)$ | $X^{2}Y-XY^{2}-2XY$ | $\operatorname{Aut}\big{(}A_{2}^{(1)}\big{)}$
$\text{P}_{\text{III}(D_{8})}$ | $(-1,2,4,1)$ | $X^{2}Y^{2}-\frac{1}{2}Y-\frac{1}{2Y}$ | $\mathbb{Z}_{4}\ltimes\mathbb{Z}_{2}$
$\text{P}_{\text{III}(D_{7})}$ | $(-1,2,3,1)$ | $X^{2}Y^{2}+X+Y$ | $\mathbb{Z}_{3}\ltimes\operatorname{Aut}\big{(}A_{1}^{(1)}\big{)}$
$\text{P}_{\text{III}(D_{6})}$ | $(0,1,2,1)$ | $X^{2}Y^{2}-XY^{2}+X$ | $\mathbb{Z}_{2}\ltimes\operatorname{Aut}\big{(}(2A_{1})^{(1)}\big{)}$
$\text{P}_{\text{V}}$ | $(1,0,1,1)$ | $X^{2}Y^{2}-X^{2}Y+XY^{2}-XY$ | $\operatorname{Aut}\big{(}A_{3}^{(1)}\big{)}$
Table 4: Hamiltonian functions of the autonomous limit defined on the set
$\\{\varepsilon_{3}=0\\}$. $(X_{3},Y_{3})$ is denoted by $(X,Y)$ for
simplicity.

In the rest of this paper, the other Painlevé equations
$(\text{P}_{\text{III}(D_{7})})$, $(\text{P}_{\text{III}(D_{8})})$,
$(\text{P}_{\text{V}})$ and $(\text{P}_{\text{VI}})$ are studied with the aid
of the weighted projective spaces and dynamical systems theory (see [3] for
$(\text{P}_{\text{I}})$, $(\text{P}_{\text{II}})$ and
$(\text{P}_{\text{IV}})$). Since the strategy is completely the same as that
for $(\text{P}_{\text{III}(D_{6})})$, we only show important steps and
formulae.

##  4 The third Painlevé equation of type $\boldsymbol{D_{7}}$

The space $\mathbb{C}P^{3}(-1{,}2{,}3{,}1)$ for
$(\text{P}_{\text{III}(D_{7})})$ is covered by four inhomogeneous coordinates
$(Y_{1}{,}Z_{1}{,}\varepsilon_{1})$, $(X_{2},Z_{2},\varepsilon_{2})$,
$(X_{3},Y_{3},\varepsilon_{3})$ and $(x,y,z)$ related as

$\displaystyle
x=\varepsilon_{1}=X_{2}\varepsilon_{2}=X_{3}\varepsilon_{3},\qquad
y=Y_{1}\varepsilon_{1}^{-2}=\varepsilon_{2}^{-2}=Y_{3}\varepsilon_{3}^{-2},\qquad
z=Z_{1}\varepsilon_{1}^{-3}=Z_{2}\varepsilon_{2}^{-3}=\varepsilon_{3}^{-3}.$

We give the third Painlevé equation of type $D_{7}$ on the local chart
$(x,y,z)$. On the other local charts, $(\text{P}_{\text{III}(D_{7})})$ is
expressed as rational differential equations. By rewriting them as
$3$-dimensional autonomous vector fields, we obtain

$\displaystyle\dot{Y}_{1}=2Y_{1}+2Y_{1}^{2}-Z_{1}+\alpha\varepsilon_{1}Y_{1},$
$\displaystyle\dot{Z}_{1}=3Z_{1}+6Y_{1}Z_{1}-\varepsilon_{1}Z_{1}+3\alpha\varepsilon_{1}Z_{1},$
(4.1)
$\displaystyle\dot{\varepsilon}_{1}=\varepsilon_{1}(1+2Y_{1}+\alpha\varepsilon_{1}),$
$\displaystyle\dot{X}_{2}=2+2X_{2}^{2}-X_{2}Z_{2}+\alpha\varepsilon_{2}X_{2},$
$\displaystyle\dot{Z}_{2}=6X_{2}Z_{2}+3Z_{2}^{2}+3\alpha\varepsilon_{2}Z_{2}-2\varepsilon_{2}Z_{2},$
$\displaystyle\dot{\varepsilon}_{2}=\varepsilon_{2}(2X_{2}+Z_{2}+\alpha\varepsilon_{2}),$

and

$\displaystyle\dot{X}_{3}=-1-2X_{3}^{2}Y_{3}+\varepsilon_{3}\left(\frac{1}{3}X_{3}-\alpha
X_{3}\right),$
$\displaystyle\dot{Y}_{3}=1+2X_{3}Y_{3}^{2}+\varepsilon_{3}\left(-\frac{2}{3}Y_{3}+\alpha
Y_{3}\right),$ (4.2)
$\displaystyle\dot{\varepsilon}_{3}=-\frac{1}{3}\varepsilon_{3}^{2}.$

We have the decomposition

$\displaystyle\mathbb{C}P^{3}(-1,2,3,1)=\mathbb{C}P^{2}(-1,2,3)\cup\mathbb{C}^{3}\qquad(\text{disjoint}),$

where $\mathbb{C}^{3}=\\{(x,y,z)\\}$ and
$\mathbb{C}P^{2}(-1,2,3)=\\{\varepsilon_{1}=0\\}\cup\\{\varepsilon_{2}=0\\}\cup\\{\varepsilon_{3}=0\\}$
in coordinates. This implies that the set $\mathbb{C}P^{2}(-1,2,3)$ is
attached at infinity of the phase space $\mathbb{C}^{3}$. Thus, the asymptotic
behavior of solutions can be studied by the limit $\varepsilon_{1}\to 0$ or
$\varepsilon_{2}\to 0$ or $\varepsilon_{3}\to 0$.

The autonomous limit is a Hamiltonian system obtained by putting
$\varepsilon_{3}=0$ for equation (4.2). On the set $\varepsilon_{3}=0$, the
action of the extended affine Weyl group
$\widetilde{W}(A_{1}^{(1)})\simeq\operatorname{Aut}(A_{1}^{(1)})\ltimes
W(A_{1}^{(1)})$ is reduced to the action of
$\operatorname{Aut}(A_{1}^{(1)})\simeq\mathbb{Z}_{2}$ defined by
$(X_{3},Y_{3})\mapsto(Y_{3},X_{3})$. Further, the autonomous limit is
invariant under the $\mathbb{Z}_{3}$ action ($r=3$) induced by the orbifold
structure, see Table 4.

The vector field (4.1) has fixed points on the infinity set
$\mathbb{C}P^{2}(-1,2,3)$ given by $(Y_{1},Z_{1},\varepsilon_{1})=(-1,0,0)$
and $(-1/2,-1/2,0)$. The Jacobi matrices of (4.1) at the fixed points are

$\displaystyle-\left(\begin{matrix}2&1&\alpha\\\ 0&3&0\\\
0&0&1\end{matrix}\right),\quad-\left(\begin{matrix}0&1&\alpha/2\\\
3&0&(3\alpha-1)/2\\\ 0&0&0\end{matrix}\right),$

respectively. The latter fixed point having a zero eigenvalue corresponds to
the irregular singular point, and the former one corresponds to a movable
singularity associated with the regular Laurent series solution (i) given in
Section 2.2; The Laurent series (i) converges to the point
$(Y_{1},Z_{1},\varepsilon_{1})=(-1,0,0)$ as $z\to z_{0}$. The exceptional
Laurent series solution (ii) does not converge to some point on
$\mathbb{C}P^{3}(-1,2,3,1)$ as $z\to z_{0}$.

######  Remark 4.1.

These fixed points are also included in the chart
$(X_{2},Z_{2},\varepsilon_{2})$. However, it is better to use the first chart
$(Y_{1},Z_{1},\varepsilon_{1})$ because the second chart has to be divided by
the $\mathbb{Z}_{2}$ action due to the orbifold structure.

To treat the Laurent series (ii), we use the Bäcklund transformation $\sigma$
defined by

$\displaystyle(\widetilde{x},\widetilde{y},\widetilde{z},\widetilde{\alpha})=\sigma(x,y,z,\alpha)=\left(-\frac{y}{z},zx,-z,1-\alpha\right).$
(4.3)

We consider another space $\mathbb{C}P^{3}(-1,2,3,1)$ with inhomogeneous
coordinates denoted by $(\widetilde{x},\widetilde{y},\widetilde{z})$ etc. We
glue two copies of $\mathbb{C}P^{3}(-1,2,3,1)$ by the transformation $\sigma$.
Then, it is easy to show that the exceptional Laurent series (ii) is converted
to the regular series (i) in the
$(\widetilde{x},\widetilde{y},\widetilde{z})$-chart, and it approaches to the
fixed point
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(-1,0,0)$
as $z\to z_{0}$.

To construct the space of initial conditions of
$(\text{P}_{\text{III}(D_{7})})$, we perform the weighted blow-ups at the
points $(Y_{1},Z_{1},\varepsilon_{1})=(-1,0,0)$ and
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(-1,0,0)$.

(i) blow-up at $(Y_{1},Z_{1},\varepsilon_{1})=(-1,0,0)$.

For the vector field (4.1), put $\hat{Y}_{1}=Y_{1}+1$ and

$\displaystyle u=\hat{Y}_{1}-Z_{1}+\alpha\varepsilon_{1},\qquad v=Z_{1},\qquad
w=\varepsilon_{2}.$

Then, the linear part is diagonalized around $(-1,0,0)$ and we obtain

$\displaystyle\dot{u}=2u-2u^{2}+2uv+4v^{2}-vw+\alpha(uw-2vw),$
$\displaystyle\dot{v}=3v+v(w-6u-6v+3\alpha w),$
$\displaystyle\dot{w}=w-w(2u+2v-\alpha w).$

The origin $(u,v,w)=(0,0,0)$ is a singularity of the foliation of integral
curves. To resolve it, we introduce the weighted blow-up defined by

$\displaystyle u=u_{1}^{2}=v_{2}^{2}u_{2}=w_{3}^{2}u_{3},\qquad
v=u_{1}^{3}v_{1}=v_{2}^{3}=w_{3}^{3}v_{3},\qquad
w=u_{1}w_{1}=v_{2}w_{2}=w_{3}.$

The weight $(2,3,1)$ is taken from the eigenvalues of the Jacobi matrix at the
singularity. The relation between the original coordinates $(x,y,z)$ and the
new coordinates $(u_{3},v_{3},w_{3})$ is given by

$\displaystyle x=w_{3},\qquad$ $\displaystyle
y=u_{3}+v_{3}w_{3}-\alpha/w_{3}-1/w_{3}^{2},\qquad$ $\displaystyle z=v_{3},$
$\displaystyle u_{3}=y-xz+\alpha/x+1/x^{2},\qquad$ $\displaystyle
w_{3}=x,\qquad$ $\displaystyle v_{3}=z.$ (4.4)

In the new coordinates, $(\text{P}_{\text{III}(D_{7})})$ is transformed into
the Hamiltonian system

$\displaystyle\frac{dw_{3}}{dz}=-\frac{\partial H_{1}}{\partial
u_{3}},\qquad\frac{du_{3}}{dz}=\frac{\partial H_{1}}{\partial w_{3}},$

with the polynomial Hamiltonian function

$\displaystyle
zH_{1}=-u+u^{2}w^{2}-\frac{1}{2}w^{2}z+2uw^{3}z+w^{4}z^{2}-\alpha\big{(}uw+w^{2}z\big{)}$
(4.5)

(here, the subscript is omitted for simplicity). Furthermore, we can verify
the symplectic relation (3.7).

(ii) blow-up at
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(-1,0,0)$.

This singularity is resolved by the same way as above by the weighted blow-up
with the weight $(2,3,1)$. The result is easily obtained by the Bäcklund
transformation $\sigma$ as follows. Define the new coordinates
$(u_{6},v_{6},w_{6})$ by

$\displaystyle
u_{6}=\widetilde{y}-\widetilde{x}\widetilde{z}+\widetilde{\alpha}/\widetilde{x}+1/\widetilde{x}^{2},\qquad
w_{6}=\widetilde{x},\qquad v_{6}=\widetilde{z}.$

Substituting (4.3) yields

$\displaystyle u_{6}=xz+xy-(1-\alpha)z/y+z^{2}/y^{2},\qquad w_{6}=-y/z,\qquad
v_{6}=-z.$ (4.6)

By this coordinates change, $(\text{P}_{\text{III}(D_{7})})$ is transformed
into the Hamiltonian system of $(u_{6},w_{6})$, whose Hamiltonian function has
the same form as (4.5), for which $\alpha$ is replaced by
$\widetilde{\alpha}=1-\alpha$.

In this manner, all singularities are resolved and the space of initial
conditions of $(\text{P}_{\text{III}(D_{7})})$ is given by
$\mathbb{C}^{2}_{(x,y)}\cup\mathbb{C}^{2}_{(u_{3},w_{3})}\cup\mathbb{C}^{2}_{(u_{6},w_{6})}$
glued by the symplectic transformations (4.4), (4.6).

##  5 The third Painlevé equation of type $\boldsymbol{D_{8}}$

The orbifold $\mathbb{C}P^{3}(-1,2,4,1)$ for $(\text{P}_{\text{III}(D_{8})})$
is covered by four inhomogeneous coordinates related as

$\displaystyle
x=\varepsilon_{1}=X_{2}\varepsilon_{2}=X_{3}\varepsilon_{3},\qquad
y=Y_{1}\varepsilon_{1}^{-2}=\varepsilon_{2}^{-2}=Y_{3}\varepsilon_{3}^{-2},\qquad
z=Z_{1}\varepsilon_{1}^{-4}=Z_{2}\varepsilon_{2}^{-4}=\varepsilon_{3}^{-4}.$

We give the third Painlevé equation of type $D_{8}$ on the local chart
$(x,y,z)$. On the other local charts, $(\text{P}_{\text{III}(D_{8})})$ is
expressed as rational differential equations. By rewriting them as
$3$-dimensional autonomous vector fields, we obtain

$\displaystyle\dot{Y}_{1}=Y_{1}^{3}-2Y_{1}^{4}-Y_{1}Z_{1},$
$\displaystyle\dot{Z}_{1}=2Y_{1}^{2}Z_{1}-2Z_{1}^{2}-8Y_{1}^{3}Z_{1}+\varepsilon_{1}Y_{1}^{2}Z_{1},$
(5.1)
$\displaystyle\dot{\varepsilon}_{1}=\varepsilon_{1}\left(\frac{1}{2}Y_{1}^{2}-2Y_{1}^{3}-\frac{1}{2}Z_{1}\right),$
$\displaystyle\dot{X}_{3}=-2X_{3}^{2}Y_{3}+\frac{1}{2}-\frac{1}{2Y_{3}^{2}}+\frac{1}{4}\varepsilon_{3}X_{3},$
$\displaystyle\dot{Y}_{3}=2X_{3}Y_{3}^{2}-\frac{1}{2}\varepsilon_{3}Y_{3},$
(5.2) $\displaystyle\dot{\varepsilon}_{3}=-\frac{1}{4}\varepsilon_{3}^{2}.$

We will not use the vector field written in
$(X_{2},Z_{2},\varepsilon_{2})$-chart.

The autonomous limit is a Hamiltonian system obtained by putting
$\varepsilon_{3}=0$ for equation (5.2). It is known that
$(\text{P}_{\text{III}(D_{8})})$ is invariant under the transformation $\pi$
defined by

$\displaystyle(\widetilde{x},\widetilde{y},\widetilde{z})=\pi(x,y,z)=\left(-\frac{xy^{2}}{z}+\frac{y}{2z},\frac{z}{y},z\right).$
(5.3)

On the set $\varepsilon_{3}=0$, this action is reduced to the $\mathbb{Z}_{2}$
action given by $(X_{3},Y_{3})\mapsto(-X_{3}Y_{3}^{2},1/Y_{3})$. Further, the
autonomous limit is invariant under the $\mathbb{Z}_{4}$ action ($r=4$)
induced by the orbifold structure, see Table 4.

The vector field (5.1) has the fixed point
$(Y_{1},Z_{1},\varepsilon_{1})=(1/2,0,0)$. The Jacobi matrix of (5.1) at the
fixed point is

$\displaystyle-\frac{1}{8}\left(\begin{matrix}2&4&0\\\ 0&4&0\\\
0&0&1\end{matrix}\right).$

The regular Laurent series solution (i) given in Section 2.2 converges to this
point as $z\to z_{0}$, while the exceptional Laurent series solution (ii) does
not converge to some point on $\mathbb{C}P^{3}(-1,2,4,1)$. To treat the
Laurent series (ii), consider another space $\mathbb{C}P^{3}(-1,2,4,1)$ with
inhomogeneous coordinates denoted by
$(\widetilde{x},\widetilde{y},\widetilde{z})$ etc. We glue two copies of
$\mathbb{C}P^{3}(-1,2,4,1)$ by the transformation (5.3). Then, it is easy to
show that the exceptional Laurent series (ii) is converted to the regular
series (i) in the $(\widetilde{x},\widetilde{y},\widetilde{z})$-chart, and it
approaches to the fixed point
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(1/2,0,0)$
as $z\to z_{0}$.

To construct the space of initial conditions of
$(\text{P}_{\text{III}(D_{8})})$, we perform the weighted blow-ups at the
points $(Y_{1},Z_{1},\varepsilon_{1})=(1/2,0,0)$ and
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(1/2,0,0)$.

(i) blow-up at $(Y_{1},Z_{1},\varepsilon_{1})=(1/2,0,0)$.

For the vector field (5.1), put $\hat{Y}_{1}=Y_{1}-1/2$ and

$\displaystyle u=\hat{Y}_{1}-2Z_{1},\qquad v=Z_{1},\qquad w=\varepsilon_{2}.$

Then, the linear part is diagonalized around $(-1,0,0)$. To resolve the
singularity $(u,v,w)=(0,0,0)$ of the resultant equation, we introduce the
weighted blow-up defined by

$\displaystyle u=u_{1}^{2}=v_{2}^{2}u_{2}=w_{3}^{2}u_{3},\qquad
v=u_{1}^{4}v_{1}=v_{2}^{4}=w_{3}^{4}v_{3},\qquad
w=u_{1}w_{1}=v_{2}w_{2}=w_{3}.$

The relation between the original coordinates $(x,y,z)$ and the new
coordinates $(u_{3},v_{3},w_{3})$ is given by

$\displaystyle x=w_{3},\qquad$ $\displaystyle
y=u_{3}+2v_{3}w_{3}^{2}+\frac{1}{2w_{3}^{2}},\qquad$ $\displaystyle z=v_{3},$
$\displaystyle u_{3}=y-2x^{2}z-\frac{1}{2x^{2}},\qquad$ $\displaystyle
w_{3}=x,\qquad$ $\displaystyle v_{3}=z.$

In the new coordinates, $(\text{P}_{\text{III}(D_{8})})$ is transformed into
the Hamiltonian system with the Hamiltonian function

$\displaystyle
zH_{1}=\frac{u}{2}+u^{2}w^{2}+w^{2}z-\frac{2}{3}w^{3}z+4uw^{4}z+4w^{6}z^{2}-\frac{w^{2}z}{1+2uw^{2}+4w^{4}z}$

(here, the subscript is omitted for simplicity). Furthermore, we can verify
the symplectic relation (3.7).

The blow-up at
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(-1,0,0)$
is done in the same way and the Hamiltonian function is easily obtained by
applying the transformation (5.3) to the above $H_{1}$. In this manner, we can
obtain the space of initial conditions.

##  6 The fifth Painlevé equation

The orbifold $\mathbb{C}P^{3}(1,0,1,1)$ for $(\text{P}_{\text{V}})$ is covered
by three inhomogeneous coordinates $(Y_{1},Z_{1},\varepsilon_{1})$,
$(X_{3},Y_{3},\varepsilon_{3})$ and $(x,y,z)$ related as

$\displaystyle x=\varepsilon_{1}^{-1}=X_{3}\varepsilon_{3}^{-1},\qquad
y=Y_{1}=Y_{3},\qquad z=Z_{1}\varepsilon_{1}^{-1}=\varepsilon_{3}^{-1}.$

The second chart does not appear because $q=0$. We give the fifth Painlevé
equation on the local chart $(x,y,z)$. On the other local charts,
$(\text{P}_{\text{V}})$ is expressed as rational differential equations. By
rewriting them as $3$-dimensional autonomous vector fields, we obtain

$\displaystyle\dot{Y}_{1}=-2Y_{1}+2Y_{1}^{2}-Y_{1}Z_{1}+Y_{1}^{2}Z_{1}+\alpha_{1}\varepsilon_{1}-(\alpha_{1}+\alpha_{3})Y_{1}\varepsilon_{1},$
$\displaystyle\dot{Z}_{1}=-Z_{1}+2Y_{1}Z_{1}-Z_{1}^{2}+\varepsilon_{1}Z_{1}+2Y_{1}Z_{1}^{2}-(\alpha_{1}+\alpha_{3})Z_{1}\varepsilon_{1}+\alpha_{2}\varepsilon_{1}Z_{1}^{2},$
(6.1)
$\displaystyle\dot{\varepsilon}_{1}=\varepsilon_{1}(-1+2Y_{1}-Z_{1}+2Y_{1}Z_{1}-(\alpha_{1}+\alpha_{3})\varepsilon_{1}+\alpha_{2}Z_{1}\varepsilon_{1}),$
$\displaystyle\dot{X}_{3}=X_{3}+X_{3}^{2}-2X_{3}Y_{3}-2X_{3}^{2}Y_{3}-\alpha_{2}\varepsilon_{3}+(\alpha_{1}+\alpha_{3}-1)\varepsilon_{3}X_{3},$
$\displaystyle\dot{Y}_{3}=-Y_{3}+Y_{3}^{2}-2X_{3}Y_{3}+2X_{3}Y_{3}^{2}+\alpha_{1}\varepsilon_{3}-(\alpha_{1}+\alpha_{3})\varepsilon_{3}Y_{3},$
(6.2) $\displaystyle\dot{\varepsilon}_{3}=-\varepsilon_{3}^{2}.$

The autonomous limit is a Hamiltonian system obtained by putting
$\varepsilon_{3}=0$ for equation (6.2). On the set $\\{\varepsilon_{3}=0\\}$,
the action of the extended affine Weyl group
$\widetilde{W}(A_{3}^{(1)})\simeq\operatorname{Aut}(A_{3}^{(1)})\ltimes
W(A_{3}^{(1)})$ is reduced to the action of $\operatorname{Aut}(A_{3}^{(1)})$
generated by $(X_{3},Y_{3})\mapsto(Y_{3}-1,-X_{3})$ and
$(X_{3},Y_{3})\mapsto(X_{3},1-Y_{3})$.

The vector field (6.1) has fixed points on the infinity set
$\mathbb{C}P^{2}(1,0,1)$ given by

$\displaystyle(Y_{1},Z_{1},\varepsilon_{1})=(0,0,0),\ (1,0,0),\ (0,-1,0),\
(1/2,-2,0),\ (1,-1,0).$

The Jacobi matrices of (6.1) at these fixed points are

$\displaystyle-\left(\begin{matrix}2&0&-\alpha_{1}\\\ 0&1&0\\\
0&0&1\end{matrix}\right),\ \left(\begin{matrix}2&0&-\alpha_{3}\\\ 0&1&0\\\
0&0&1\end{matrix}\right),\ \left(\begin{matrix}-1&0&\alpha_{1}\\\ 0&1&*\\\
0&0&0\end{matrix}\right),\ \left(\begin{matrix}0&-1/4&*\\\ 4&0&*\\\
0&0&0\end{matrix}\right),\ \left(\begin{matrix}1&0&-\alpha_{3}\\\ 0&-1&*\\\
0&0&0\end{matrix}\right),$

respectively, where $*$ denotes certain long numerical expressions. The latter
three fixed points having zero eigenvalues correspond to the irregular
singular point. The center-(un)stable manifolds at these points can be exactly
calculated for certain specific values of parameters, which give the Riccati
equations of confluent hypergeometric type (Table 3). The former two points
correspond to movable singularities associated with the regular Laurent series
solutions (ii) and (i) given in Section 2.2. The Laurent series (ii) and (i)
converge to the points $(Y_{1},Z_{1},\varepsilon_{1})=(0,0,0)$ and $(1,0,0)$
as $z\to z_{0}$, respectively. The exceptional Laurent series solutions (iii)
and (iv) do not converge to some point on $\mathbb{C}P^{3}(1,0,1,1)$ as $z\to
z_{0}$.

To treat the Laurent series (iii) and (iv), we use the Bäcklund transformation
$\pi$ defined by

$\displaystyle(\widetilde{x},\widetilde{y},\widetilde{z},\widetilde{\alpha}_{1},\widetilde{\alpha}_{2},\widetilde{\alpha}_{3})=\pi(x,y,z,\alpha_{1},\alpha_{2},\alpha_{3})=\left(z(y-1),-\frac{x}{z},z,\alpha_{2},\alpha_{3},1-\alpha_{1}-\alpha_{2}-\alpha_{3}\right).$

We consider another space $\mathbb{C}P^{3}(1,0,1,1)$ with inhomogeneous
coordinates denoted by $(\widetilde{x},\widetilde{y},\widetilde{z})$ etc. We
glue two copies of $\mathbb{C}P^{3}(1,0,1,1)$ by the transformation $\pi$.
Then, the exceptional Laurent series (iii) and (iv) are converted to the
regular series (i) and (ii), respectively, in the
$(\widetilde{x},\widetilde{y},\widetilde{z})$-chart. They converge to the
fixed points
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(1,0,0)$
and $(0,0,0)$ as $z\to z_{0}$, respectively.

To construct the space of initial conditions of $(\text{P}_{\text{V}})$, we
perform the weighted blow-ups at these four points.

(i) blow-up at $(Y_{1},Z_{1},\varepsilon_{1})=(0,0,0)$.

For the vector field (6.1), put $u=Y_{1}-\alpha_{1}\varepsilon_{1}$,
$v=Z_{1}$, $w=\varepsilon_{1}$. Then, the linear part is diagonalized. For the
system of $(u,v,w)$, we introduce the weighted blow-up defined by

$\displaystyle u=u_{1}^{2}=v_{2}^{2}u_{2}=w_{3}^{2}u_{3},\qquad
v=u_{1}v_{1}=v_{2}=w_{3}v_{3},\qquad w=u_{1}w_{1}=v_{2}w_{2}=w_{3}.$

The weight $(2,1,1)$ is taken from the eigenvalues of the Jacobi matrix at the
singularity. The relation between the original coordinates $(x,y,z)$ and the
new coordinates $(u_{3},v_{3},w_{3})$ is given by

$\displaystyle x=1/w_{3},\qquad$ $\displaystyle
y=u_{3}w_{3}^{2}+\alpha_{1}w_{3},\qquad$ $\displaystyle z=v_{3},$
$\displaystyle u_{3}=x^{2}y-\alpha_{1}x,\qquad$ $\displaystyle
w_{3}=1/x,\qquad$ $\displaystyle v_{3}=z.$

In the new coordinates, $(\text{P}_{\text{V}})$ is transformed into the
Hamiltonian system, whose Hamiltonian function is

$\displaystyle
zH_{1}=u-u^{2}w^{2}-u^{2}w^{3}z+(z-\alpha_{1}+\alpha_{3})uw-(2\alpha_{1}+\alpha_{2})uw^{2}z-\alpha_{1}(\alpha_{1}+\alpha_{2})wz$

(here, the subscript is omitted for simplicity). Furthermore, we can verify
the symplectic relation (3.7).

(ii) blow-up at $(Y_{1},Z_{1},\varepsilon_{1})=(1,0,0)$.

For the vector field (6.1), put $\hat{Y}_{1}=Y_{1}-1$ and
$u=\hat{Y}_{1}-\alpha_{3}\varepsilon_{1}$, $v=Z_{1}$, $w=\varepsilon_{1}$.
Then, the linear part is diagonalized around $(1,0,0)$. For the system of
$(u,v,w)$, we introduce the weighted blow-up defined by

$\displaystyle u=u_{4}^{2}=v_{5}^{2}u_{5}=w_{6}^{2}u_{6},\qquad
v=u_{4}v_{4}=v_{5}=w_{6}v_{6},\qquad w=u_{4}w_{4}=v_{5}w_{5}=w_{6}.$

The relation between the original coordinates $(x,y,z)$ and the coordinates
$(u_{6},v_{6},w_{6})$ is given by

$\displaystyle x=1/w_{6},\qquad$ $\displaystyle
y=u_{6}w_{6}^{2}+\alpha_{3}w_{6}+1,\qquad$ $\displaystyle z=v_{6},$
$\displaystyle u_{6}=x^{2}y-x^{2}-\alpha_{3}x,\qquad$ $\displaystyle
w_{6}=1/x,\qquad$ $\displaystyle v_{6}=z.$

In the new coordinates, $(\text{P}_{\text{V}})$ is transformed into the
Hamiltonian system, whose Hamiltonian function is

$\displaystyle
zH_{1}=-u-u^{2}w^{2}-u^{2}w^{3}z+(-z+\alpha_{1}-\alpha_{3})uw-(2\alpha_{3}+\alpha_{2})uw^{2}z-\alpha_{3}(\alpha_{2}+\alpha_{3})wz$

(here, the subscript is omitted for simplicity). Again, we can verify the
symplectic relation (3.7).

The singularities
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(1,0,0)$
and $(0,0,0)$ are resolved by the same way as above by the weighted blow-up
with the weight $(2,1,1)$. The result is easily obtained by the Bäcklund
transformation $\pi$ as in the previous sections. In this manner, all
singularities are resolved and it turns out that the space of initial
conditions of $(\text{P}_{\text{V}})$ is given by five copies of
$\mathbb{C}^{2}$ glued by the symplectic transformations.

##  7 The sixth Painlevé equation

The orbifold $\mathbb{C}P^{3}(1,0,0,1)$ for $(\text{P}_{\text{VI}})$ is
covered by two inhomogeneous coordinates $(Y_{1},Z_{1},\varepsilon_{1})$ and
$(x,y,z)$ related as

$\displaystyle x=\varepsilon_{1}^{-1},\qquad y=Y_{1},\qquad z=Z_{1}.$

We give the sixth Painlevé equation on the local chart $(x,y,z)$. On the other
local chart $(Y_{1},Z_{1},\varepsilon_{1})$, $(\text{P}_{\text{VI}})$ is
expressed as a rational differential equation. As before, we rewrite it as a
$3$-dimensional autonomous polynomial vector field, whose expression is too
long and omitted here. This vector field has fixed points

$\displaystyle(Y_{1},Z_{1},\varepsilon_{1})=(0,c,0),\ (1,c,0),\ (c,c,0),$

where $c\in\mathbb{C}$ is an arbitrary constant. The Jacobi matrices at these
fixed points are

$\displaystyle c\left(\begin{matrix}2&0&-\alpha_{4}\\\ 0&0&c-1\\\
0&0&1\end{matrix}\right),\quad(1-c)\left(\begin{matrix}2&0&-\alpha_{3}\\\
0&0&-c\\\ 0&0&1\end{matrix}\right),\quad
c(c-1)\left(\begin{matrix}2&-2&1-\alpha_{0}\\\ 0&0&1\\\
0&0&1\end{matrix}\right),$

respectively. These fixed points correspond to movable singularities
associated with the regular Laurent series solutions (iii), (ii) and (i) given
in Section 2.2. The Laurent series (iii), (ii) and (i) converge to these
points as $z\to z_{0}$, respectively. The exceptional Laurent series solutions
(iv) and (v) do not converge to some point on $\mathbb{C}P^{3}(1,0,0,1)$ as
$z\to z_{0}$.

To treat the Laurent series (iv) and (v), we use the Bäcklund transformation
$\pi_{1}$ defined by

$\displaystyle(\widetilde{x},\widetilde{y},\widetilde{z},\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\widetilde{\alpha}_{2},\widetilde{\alpha}_{3},\widetilde{\alpha}_{4})=\pi_{1}(x,y,z,\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$
$\displaystyle\hphantom{(\widetilde{x},\widetilde{y},\widetilde{z},\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\widetilde{\alpha}_{2},\widetilde{\alpha}_{3},\widetilde{\alpha}_{4})}{}=\left(-\frac{y}{z}(xy+\alpha_{2}),\frac{z}{y},z,\alpha_{3},\alpha_{4},\alpha_{2},\alpha_{0},\alpha_{1}\right).$

We consider another space $\mathbb{C}P^{3}(1,0,0,1)$ with inhomogeneous
coordinates denoted by $(\widetilde{x},\widetilde{y},\widetilde{z})$ etc. We
glue two copies of $\mathbb{C}P^{3}(1,0,0,1)$ by the transformation $\pi_{1}$.
Then, the exceptional Laurent series (iv) becomes the regular series (iii) in
the $(\widetilde{x},\widetilde{y},\widetilde{z})$-chart. It converges to the
fixed point
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(0,c,0)$ as
$z\to z_{0}$. On the other hand, the exceptional Laurent series (v) is
expressed as $\widetilde{x}(z)\sim O(1)$, $\widetilde{y}(z)\sim O(T)$ in the
$(\widetilde{x},\widetilde{y},\widetilde{z})$-chart. Since the solution is
holomorphic at $z=z_{0}$, we need not resolve the singularity.

To construct the space of initial conditions of $(\text{P}_{\text{VI}})$, we
perform the weighted blow-ups at four points.

(i) blow-up at $(Y_{1},Z_{1},\varepsilon_{1})=(0,c,0)$.

For the vector field written in $(Y_{1},Z_{1},\varepsilon_{1})$-chart, put
$u=Y_{1}-\alpha_{4}\varepsilon_{1}$, $v=Z_{1}$, $w=\varepsilon_{1}$. Then, we
introduce the weighted blow-up with the weight $(2,0,1)$ defined by

$\displaystyle u=u_{1}^{2}=w_{2}^{2}u_{2},\qquad v=v_{1}=v_{2},\qquad
w=u_{1}w_{1}=w_{2}.$

This is a blow-up along the line $\\{v=\text{const}\\}$. The relation between
the original coordinates $(x,y,z)$ and the new coordinates
$(u_{2},v_{2},w_{2})$ is given by

$\displaystyle x=1/w_{2},\qquad$ $\displaystyle
y=u_{2}w_{2}^{2}+\alpha_{2}w_{2},\qquad$ $\displaystyle z=v_{2},$
$\displaystyle u_{2}=x^{2}y-\alpha_{2}x,\qquad$ $\displaystyle
w_{2}=1/x,\qquad$ $\displaystyle v_{2}=z.$

In the new coordinates, $(\text{P}_{\text{VI}})$ is transformed into the
Hamiltonian system, whose Hamiltonian function is

$\displaystyle
z(z-1)H_{1}=u^{3}w^{4}+uz-u^{2}w^{2}(1+z)+(1-\alpha_{0}-\alpha_{3}+2\alpha_{4})u^{2}w^{3}$
$\displaystyle\hphantom{z(z-1)H_{1}=}{}+(\alpha_{0}-\alpha_{4}+z\alpha_{3}-z\alpha_{4}-1)uw+\alpha_{4}\big{(}\alpha_{1}\alpha_{2}+\alpha_{2}^{2}+(1-\alpha_{0}-\alpha_{3})\alpha_{4}\big{)}w$
$\displaystyle\hphantom{z(z-1)H_{1}=}{}+\big{(}\alpha_{1}\alpha_{2}+\alpha_{2}^{2}+2\alpha_{4}-2\alpha_{0}\alpha_{4}-2\alpha_{3}\alpha_{4}+\alpha_{4}^{2}\big{)}uw^{2}$

(here, the subscript is omitted for simplicity). Furthermore, we can verify
the symplectic relation (3.7).

The other singularities $(Y_{1},Z_{1},\varepsilon_{1})=(1,c,0),(c,c,0)$ and
$(\widetilde{Y}_{1},\widetilde{Z}_{1},\widetilde{\varepsilon}_{1})=(0,c,0)$
are resolved by the same way as above by the weighted blow-up with the weight
$(2,0,1)$. In this manner, all singularities are resolved and the space of
initial conditions of $(\text{P}_{\text{VI}})$ is obtained by six copies of
$\mathbb{C}^{2}$ glued by the symplectic transformations; i.e., $(x,y)$,
$(\widetilde{x},\widetilde{y})$ and four charts arising from blow-ups at four
points. We need $(\widetilde{x},\widetilde{y})$-chart for the exceptional
Laurent series (v).

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