  We consider a class of quasi-integrable Hamiltonian systems obtained by
adding to a non-convex Hamiltonian function of an integrable system a
perturbation depending only on the angle variables. We focus on a resonant
maximal torus of the unperturbed system, foliated into a family of
lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow
with rotation vector satisfying some mild Diophantine condition. We show that
at least one lower-dimensional torus with that rotation vector always exists
also for the perturbed system. The proof is based on multiscale analysis and
resummation procedures of divergent series. A crucial role is played by
suitable symmetries and cancellations, ultimately due to the Hamiltonian
structure of the system.
