The Third, Fifth and Sixth Painlev\'e Equations on Weighted Projective Spaces

The third, fifth and sixth Painlev\'e 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.

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 dx i dz = f i (x 1 , . . . , x m , z), i = 1, . . . , m. (1.1) The exponent of a monomial x µ 1 1 · · · x µm m z η included in the right hand side f i is defined as (µ 1 , . . . , µ i−1 , µ i − 1, µ i+1 , . . . , µ m , η + 1). Each exponent specifies a point of the integer lattice in R m+1 . The Newton polyhedron of (1.1) is the convex hull of the union of the positive quadrants 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.
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 (A2) any monomials x µ 1 1 · · · x µm m z η included in g i satisfy p 1 µ 1 + · · · + p i (µ i − 1) + · · · + p m µ m + r(η + 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 Z s -action given by (x 1 , . . . , x m , z) → ω p 1 x 1 , . . . , ω pm x m , ω r z , ω := e 2πi/s . A tuple of positive integers (p 1 , . . . , 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 +· · ·+p m x m +rx = s (assumption (A1)), then the associated toric variety is the weighted projective space CP m+1 (p 1 , . . . , p m r, s), which is an (m + 1)-dimensional orbifold, see Section 2.1 for the definition.
The orbifold CP m+1 (p 1 , . . . , p m , r, s) is regarded as a compactification of the phase space C m+1 = {(x 1 , . . . , 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 CP m+1 (p 1 , . . . , 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).
In this paper, the third, fifth and sixth Painlevé equations are studied with the aid of the weighted projective spaces CP 3 (p, q, r, s) and dynamical systems theory. The weighted projective space is decomposed into the disjoint sum CP 3 (p, q, r, s) CP 2 (p, q, r) ∪ C 3 . This implies that the natural phase space C 3 = {(x, y, z)} of the Painlevé equations is embedded in CP 3 (p, q, r, s) and the two-dimensional manifold CP 2 (p, q, r) is attached at infinity. We regard the Painlevé equation as a three-dimensional autonomous vector field defined on CP 3 (p, q, r, s). Then, the vector field has several fixed points on the infinity set CP 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 CP 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 (P III(D 7 ) ), (P III(D 8 ) ), (P V ) and (P VI ) is completely the same as that for (P III(D 6 ) ), we only show a sketch and several formulae for them after Section 4. See [3] for (P I ), (P II ) and (P IV ).
Due to the choice of the branch of x 1/p , we also obtain by putting x → e 2πi x. This implies that the subset of CP 3 (p, q, r, s) such that x = 0 is homeomorphic to C 3 /Z p , where the Z p -action is defined as above.

H. Chiba
This proves that the orbifold structure of CP 3 (p, q, r, s) is given by The local charts (Y 1 , Z 1 , ε 1 ), (X 2 , Z 2 , ε 2 ), (X 3 , Y 3 , ε 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 C 3 , not on the quotient C 3 /Z i (i = p, q, r, s). Therefore, any equations written in these inhomogeneous coordinates should be invariant under the corresponding 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 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 CP 3 (0, q, r, s) C × CP 2 (q, r, s). CP 3 (p, q, r, s) is compact if and only if all p, q, r, s are positive.

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 where (A 0 , B 0 ) = (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 κ such that (A κ , B κ ) 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 CP 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 κ of exceptional Laurent series solutions are well-defined and given as in Table 1. In what follows, denote T := z − z 0 . (P III(D 6 ) ): The third Painlevé equation of type D 6 has three types of Laurent series solutions given by 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 κ = 2.
(P III(D 7 ) ): The third Painlevé equation of type D 7 has two types of Laurent series solutions given by 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 κ = 2.
(P III(D 8 ) ): The third Painlevé equation of type D 8 has two types of Laurent series solutions given by 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 κ = 2.
(P V ): The fifth Painlevé equation has four types of Laurent series solutions given by 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 κ = 2.
(P VI ): The sixth Painlevé equation has five types of Laurent series solutions given by 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 κ = 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 D 6 3.1 P III(D 6 ) on CP 3 (0, 1, 2, 1) The orbifold structure of CP 3 (0, 1, 2, 1) is given by Thus, the space is covered by three inhomogeneous coordinates (X 2 , Z 2 , ε 2 ), (X 3 , Y 3 , ε 3 ) and (x, y, z) related as We give the third Painlevé equation of type D 6 on the local chart (x, y, z). On the other local charts, (P III(D 6 ) ) is expressed as The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces 9 In order to apply dynamical systems theory later, it is convenient to rewrite them as 3-dimensional autonomous vector fields of the forṁ where (˙) = d/dt and t ∈ C parameterizes each integral curve. We also have the decomposition Since CP 2 (0, 1, 2) = C × CP 1 (1, 2) and CP 1 (1, 2) is the usual projective line, This implies that the set C × CP 1 is attached at infinity of the natural phase space C 3 = {(x, y, z)} of (P III(D 6 ) ), and the asymptotic behavior of solutions can be studied by the limit ε 2 → 0 or The Jacobi matrices of (3.2) at the fixed points are The latter two points, for which Z 2 = 0 ⇒ z = ∞, 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| → ∞.
As a simple application, we can prove the next theorem. Similarly, the vector field is locally linearized around (X 2 , Z 2 , ε 2 ) = (1, 0, 0) and ( X 2 , Z 2 , ε 2 ) = (1, 0, 0). This result implies that (P 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].

The space of initial conditions
A purpose in this section is to construct the space of initial conditions for (P III(D 6 ) ). On the manifold M 1 , there are three singularities of the foliation of integral curves; (X 2 , Z 2 , ε 2 ) = (1, 0, 0), (0, 0, 0) and ( X 2 , Z 2 , ε 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, (P 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 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 , ε 2 ) = (0, 0, 0). For the vector field (3.2), put Then, the linear part is diagonalized and we obtaiṅ 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 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} ∪ {v 2 = 0} ∪ {w 3 = 0} is the weighted projective space CP 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 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, (P III(D 6 ) ) is transformed into the Hamiltonian system with the Hamiltonian function (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 (ii) blow-up at (X 2 , Z 2 , ε 2 ) = (1, 0, 0). Putting X 2 − 1 =X 2 and u =X 2 + βε 2 , v = Z 2 , w = ε 2 for the vector field (3.2) results iṅ To resolve the singularity at (u, v, w) = (0, 0, 0), we introduce the weighted blow-up with the weight (2, 2, 1) The relation between the original coordinates (x, y, z) and the new coordinates (u 6 , v 6 , w 6 ) is given by which defines a fiber bundle over z-space. In the new coordinates, (P III(D 6 ) ) is written as the Hamiltonian system with the Hamiltonian function (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).
In this manner, all singularities are resolved and we have Theorem 3.2. The space of initial conditions E(z) of (P III(D 6 ) ) is given by C 2 (x,y) ∪ C 2 (u 3 ,w 3 ) ∪ C 2 (u 6 ,w 6 ) ∪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 ∈ C\{0}, on which (P III(D 6 ) ) is expressed as a polynomial Hamiltonian system.

The Riccati solutions
It is known that when the parameter α is an integer or β is an integer, there exists a oneparameter family of solutions of (P III(D 6 ) ) satisfying the Riccati equation, which is equivalent to the Bessel equation. For example, when α = 0 (resp. β = 0), the Riccati equation is given by z dy dz = −y 2 + βy + z, resp. z dy dz = y 2 + αy + z. (3.11) The Bäcklund transformations yield the Riccati equations for a general case α ∈ Z or β ∈ 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 , ε 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.
Substituting this relation to equation (1.4) yields the Riccati equation for α = 1, though it is also obtained by the Bäcklund transformation to that for α = 0. The same argument at the point (X 2 , Z 2 , ε 2 ) = (1, −1, 0) provides the Riccati equation for β = 0. Theorem 3.3. When α ∈ Z or β ∈ Z, there exists a one-parameter family of solutions of (P 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 CP 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 (P I ), (P III(D 7 ) ) and (P III(D 8 ) ). For such specific values of parameters, center-(un)stable manifolds in CP 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 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 CP 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.

Boutroux's coordinates
For the first and second Painlevé equations, the third local chart (X 3 , Y 3 , ε 3 ) of CP 3 (p, q, r, s) defined by equation ( which is often called the autonomous limit. Since a generic integral curve given by H III(D 6 ) = X 2 3 Y 2 3 − X 3 Y 2 3 + X 3 = const is an elliptic curve, a general solution can be expressed by Weierstrass's elliptic functions. Then, the system (3.3) with small ε 3 can be studied by a perturbation method.
Let us calculate the action of the extended affine Weyl group W (2A 1 ) (1) restricted on the set {ε 3 = 0}, which leaves the autonomous limit (3.12) invariant.
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 , ε 3 )-chart. For example, the action of s 1 in the (X 3 , Y 3 , ε 3 )-chart is given by On the set {ε 3 = 0}, it is reduced to Since the autonomous limit (3.12) is independent of the parameters α, β, the action of s 1 to (3.12) is trivial. Similarly, the actions of s 0 , s 0 and s 1 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 π 1 , π 2 , σ 1 are not trivial, which are explicitly given by Furthermore, (3.12) is invariant under the Z 2 action Y 3 → −Y 3 due to the orbifold structure of CP 3 (0, 1, 2, 1). Since r = deg(z) = 2, (X 3 , Y 3 , ε 3 ) are coordinates on the lift C 3 of the quotient A similar result also holds for the other Painlevé equations except for (P 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 = Aut R (1) because the action of W R (1) on the set {ε 3 = 0} is reduced to the trivial action as above. If r > 1, the autonomous limit is further invariant under the Z r action arising from the orbifold structure of CP 3 (p, q, r, s). The autonomous limit on the set {ε 3 = 0} is not defined for (P VI ) in this way because r = 0.
weight H J symmetry P I (3, 2, 4, 5) Table 4. Hamiltonian functions of the autonomous limit defined on the set In the rest of this paper, the other Painlevé equations (P III(D 7 ) ), (P III(D 8 ) ), (P V ) and (P VI ) are studied with the aid of the weighted projective spaces and dynamical systems theory (see [3] for (P I ), (P II ) and (P IV )). Since the strategy is completely the same as that for (P III(D 6 ) ), we only show important steps and formulae.

H. Chiba
We give the third Painlevé equation of type D 7 on the local chart (x, y, z). On the other local charts, (P III(D 7 ) ) is expressed as rational differential equations. By rewriting them as 3-dimensional autonomous vector fields, we obtaiṅ We have the decomposition This implies that the set CP 2 (−1, 2, 3) is attached at infinity of the phase space C 3 . Thus, the asymptotic behavior of solutions can be studied by the limit ε 1 → 0 or ε 2 → 0 or ε 3 → 0. The autonomous limit is a Hamiltonian system obtained by putting ε 3 = 0 for equation (4.2). On the set ε 3 = 0, the action of the extended affine Weyl group W (A ). Further, the autonomous limit is invariant under the Z 3 action (r = 3) induced by the orbifold structure, see Table 4.
Remark 4.1. These fixed points are also included in the chart (X 2 , Z 2 , ε 2 ). However, it is better to use the first chart (Y 1 , Z 1 , ε 1 ) because the second chart has to be divided by the Z 2 action due to the orbifold structure.
To treat the Laurent series (ii), we use the Bäcklund transformation σ defined by We consider another space CP 3 (−1, 2, 3, 1) with inhomogeneous coordinates denoted by ( x, y, z) etc. We glue two copies of CP 3 (−1, 2, 3, 1) by the transformation σ. Then, it is easy to show that the exceptional Laurent series (ii) is converted to the regular series (i) in the ( x, y, z)-chart, and it approaches to the fixed point ( Y 1 , Z 1 , ε 1 ) = (−1, 0, 0) as z → z 0 .
(i) blow-up at (Y 1 , Z 1 , ε 1 ) = (−1, 0, 0). For the vector field (4.1), putŶ 1 = Y 1 + 1 and Then, the linear part is diagonalized around (−1, 0, 0) and we obtaiṅ 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 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 In the new coordinates, (P III(D 7 ) ) is transformed into the Hamiltonian system with the polynomial Hamiltonian function (here, the subscript is omitted for simplicity). Furthermore, we can verify the symplectic relation (3.7).
(ii) blow-up at ( Y 1 , Z 1 , ε 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 σ as follows. Define the new coordinates (u 6 , v 6 , w 6 ) by Substituting (4.3) yields By this coordinates change, (P 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 α is replaced by α = 1 − α.

The third Painlevé equation of type D 8
The orbifold CP 3 (−1, 2, 4, 1) for (P III(D 8 ) ) is covered by four inhomogeneous coordinates related as We give the third Painlevé equation of type D 8 on the local chart (x, y, z). On the other local charts, (P III(D 8 ) ) is expressed as rational differential equations. By rewriting them as 3-dimensional autonomous vector fields, we obtaiṅ We will not use the vector field written in (X 2 , Z 2 , ε 2 )-chart. The autonomous limit is a Hamiltonian system obtained by putting ε 3 = 0 for equation (5.2). It is known that (P III(D 8 ) ) is invariant under the transformation π defined by ( x, y, z) = π(x, y, z) = − xy 2 z + y 2z , z y , z .
On the set ε 3 = 0, this action is reduced to the Z 2 action given by (X 3 , Y 3 ) → (−X 3 Y 2 3 , 1/Y 3 ). Further, the autonomous limit is invariant under the Z 4 action (r = 4) induced by the orbifold structure, see Table 4.
The blow-up at ( Y 1 , Z 1 , ε 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.

The f ifth Painlevé equation
The orbifold CP 3 (1, 0, 1, 1) for (P V ) is covered by three inhomogeneous coordinates (Y 1 , Z 1 , ε 1 ), (X 3 , Y 3 , ε 3 ) and (x, y, z) related as 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, (P V ) is expressed as rational differential equations. By rewriting them as 3-dimensional autonomous vector fields, we obtaiṅ The autonomous limit is a Hamiltonian system obtained by putting ε 3 = 0 for equation (6.2). On the set {ε 3 = 0}, the action of the extended affine Weyl group W (A 3 ) Aut(A 3 ) is reduced to the action of Aut(A The vector field (6.1) has fixed points on the infinity set CP 2 (1, 0, 1) given by The Jacobi matrices of (6.1) at these fixed points are 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 , ε 1 ) = (0, 0, 0) and (1, 0, 0) as z → z 0 , respectively. The exceptional Laurent series solutions (iii) and (iv) do not converge to some point on CP 3 (1, 0, 1, 1) as z → z 0 .
To construct the space of initial conditions of (P V ), we perform the weighted blow-ups at these four points.
(i) blow-up at (Y 1 , Z 1 , ε 1 ) = (0, 0, 0). For the vector field (6.1), put u = Y 1 − α 1 ε 1 , v = Z 1 , w = ε 1 . Then, the linear part is diagonalized. For the system of (u, v, w), we introduce the weighted blow-up defined by 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 In the new coordinates, (P V ) is transformed into the Hamiltonian system, whose Hamiltonian function is (here, the subscript is omitted for simplicity). Furthermore, we can verify the symplectic relation (3.7).
In the new coordinates, (P V ) is transformed into the Hamiltonian system, whose Hamiltonian function is zH 1 = −u − u 2 w 2 − u 2 w 3 z + (−z + α 1 − α 3 )uw − (2α 3 + α 2 )uw 2 z − α 3 (α 2 + α 3 )wz (here, the subscript is omitted for simplicity). Again, we can verify the symplectic relation (3.7). The singularities ( Y 1 , Z 1 , ε 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 π as in the previous sections. In this manner, all singularities are resolved and it turns out that the space of initial conditions of (P V ) is given by five copies of C 2 glued by the symplectic transformations.
To construct the space of initial conditions of (P VI ), we perform the weighted blow-ups at four points.