Abstract
The class of so-called Henon-Heiles systems is slightly broadened by allowing for the existence of non-standard Hamiltonians. The extra parameter in the equations of motion is shown to give rise to a generalization of the three known integrability cases. In addition, three degenerate cases are detected, characterized by a partial decoupling of the equations. For these cases, the author still obtains two independent first integrals, but their involutiveness can only be understood in terms of a nonstandard Poisson structure.