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We consider a class of quasi-integrable Hamiltonian systems obtained byadding to a non-convex Hamiltonian function of an integrable system aperturbation depending only on the angle variables. We focus on a resonantmaximal torus of the unperturbed system, foliated into a family oflower-dimensional tori of codimension 1, invariant under a quasi-periodic flowwith rotation vector satisfying some mild Diophantine condition. We show thatat least one lower-dimensional torus with that rotation vector always existsalso for the perturbed system. The proof is based on multiscale analysis andresummation procedures of divergent series. A crucial role is played bysuitable symmetries and cancellations, ultimately due to the Hamiltonianstructure of the system.

Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions