Integrable Hamiltonian systems and the Painlevé property

Abstract

A direct method is described for obtaining conditions under which certain $N$-degree-of-freedom Hamiltonian systems are integrable, i.e., possess $N$ integrals in involution. This method consists of requiring that the general solutions have the Painlev\'e property, i.e., no movable singularities other than poles. We apply this method to several Hamiltonian systems of physical significance such as the generalized H\'enon-Heiles problem and the Toda lattice with $N=2 \mathrm{and} 3$, and recover all known integrable cases together with a few new ones. For some of these cases the second integral is written down explicitly while for others integrability is confirmed by numerical experiments.