On some Hamiltonian structures of Painlevé systems. II (Q1964220)

This paper gives certain patching descriptions of the spaces of initial conditions or the defining manifolds of Painlevé systems \((H_J)\), \(J= V, IV, III, II\) (equivalent to Painlevé equations \(P_J\)) which were constructed by K. Okamoto. Each initial condition space is described as patching of several complex affine planes \(\mathbb{C}^2\) by some birational symplectic transformations. The initial condition space is a disjoint union of the original complex plane and several complex lines. The original Hamiltonian system \((H_J)\) is extended to the other charts as Hamiltonian systems each Hamiltonian of which is a polynomial of the canonical variables. The uniqueness of Hamiltonian systems on the above defined manifolds is proved in the third part of this series, which has already been published [\textit{A. Matumiya}, Kumamoto J. Math. 10, 45-73 (1997; Zbl 0887.34006)]. For part I see [\textit{T. Shioda} and \textit{K. Takano}, Funkc. Ekvacioj, Ser. Int. 40, No. 2, 271-291 (1997; Zbl 0891.34003)].