Geometry of slow-fast Hamiltonian systems and Painlevé equations

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In the first part of the paper we introduce some geometric tools needed to describe slow-fast Hamiltonian systems on smooth manifolds. We start with a smooth Poisson bundle p: M→ B of a regular (i.e. of constant rank) Poisson manifold (M,ω) over a smooth symplectic manifold (B,λ), the foliation into leaves of the bundle coincides with the symplectic foliation generated by the Poisson structure on M. This defines a singular symplectic structure Ωε= ω+ ε⁻¹p*λ on M for any positive small ε, where p*λ is a lift of 2-form λ on M. Given a smooth Hamiltonian H on M one gets a slow-fast Hamiltonian system w.r.t. Ωε. We define a slow manifold SM of this system. Assuming SM to be a smooth submanifold, we define a slow Hamiltonian flow on SM. The second part of the paper deals with singularities of the restriction of p on SM and their relations with the description of the system near them. It appears, if M = 4, B = 2 and Hamilton function H is generic, then behavior of the system near singularities of the fold type is described in the principal approximation by the equation Painlevé-I, but if a singular point is a cusp, then the related equation is Painlevé-II. This fact for particular types of Hamiltonian systems with one and a half degrees of freedom was discovered earlier by R.Haberman.