Painlevé Conjecture Revisited

The discovery of new integrable two-dimensional Hamiltonian systems is reported. The analytic structure of the solutions makes necessary the generalization of the Painlev\'e conjecture, a widely used integrability criterion. Such a generalization is presented, which the authors believe should replace the usual conjecture for two-dimensional Hamiltonian systems. It is indeed compatible with all the systems already found and, in addition, leads to still new integrable cases.