On Some Hamiltonian Structures of Painlevé Systems, I

In this series of papers, we study some Hamiltonian structures of Painlevé systems (HJ ), J = V I, V, IV, III, II, I, namely, symplectic structures of the spaces for Painlevé systems constructed by K. Okamoto([7]), and a characterization of Painlevé systems by their spaces. As is well known, P. Painlevé and B. Gambier discovered, at the beginning of this century, six nonlinear second order differential equations free from movable branch points, which are now called the Painlevé equations. We denote them by PJ , J = V I, V, IV, III, II, I. For example, the sixth Painlevé equation PV I is given by dx dt2 = 1 2 ( 1 x + 1 x− 1 + 1 x− t )( dx dt )2 − ( 1 t + 1 t− 1 + 1 x− t ) dx dt