Existence of KAM Tori in Degenerate Hamiltonian Systems

This article is devoted to the proof that invariant tori still exist in perturbed integrable Hamiltonian systems with n degrees of freedom H(p, q) = N(p) + P(p, q) if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping ∂N/∂p can be any positive integer smaller than n provided the image of any open set under the mapping includes a Cn + 2 curved one-dimensional submanifold.