On Hamiltonian structures of quasi-Painlevé equations

Abstract

Abstract We describe the quasi-Painlevé property of a system of ordinary differential equations in terms of a global Hamiltonian structure on an analogue of Okamoto’s space of initial conditions for the Painlevé equations. In the quasi-Painlevé case, the Hamiltonian structure is with respect to a two-form which is allowed to have certain zeroes on the surfaces forming the space of initial conditions, as opposed to holomorphic symplectic forms in the case of the Painlevé equations. We provide the spaces and Hamiltonian structures for several known quasi-Painlevé equations and also for a new example, which we prove to have the quasi-Painlevé property via the Hamiltonian structure and construction of an appropriate auxiliary function which remains bounded on solutions.