Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form , where (x,y,z) Tn×Rn×R2m, ε is a small parameter, and M(ω) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.