Painlevé property and geometry

The Painlevé property for discrete Hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times. The augmented manifold is constructed by expanding the Hamilton-Jacobi equation. A complete classification of the types of poles allowed in complex time is given for Hamiltonians which separate into the direct product of hyperelliptic curves. For such systems, bounds on the degrees of the (polynomial) separating variable change, and the other integrals in involution can be found from the pole series and the Hamilton-Jacobi equation. It is shown how branching can arise naturally in a Painlevé system.