On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom

Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the "intermediate system" depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable "convexity" assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In "non convex" cases, stability holds up to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a "blow up" (complex) analysis near separatrices, KAM techniques and energy conservation arguments.