Explicit integration of the H\'enon-Heiles Hamiltonians

Abstract

We consider the cubic and quartic He'non-Heiles Hamiltonians with additional inverse square terms, which pass the Painleve' test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painleve' sense (impossibility to add any term to the Hamiltonian without destroying the Painleve' property).