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4. . The nondegeneracy condition requires that the set of invariant tori can be locally labeled by means of the frequencies on each torus. Another possible condition independent from the previous one is the so-called isoenergetic nondegeneracy condition which considers the restriction of the Hamiltonian on the surface of constant energy. Very often realistic models of physical systems are perturbations of integrable systems that are degenerate and also isoenergetically degenerate (such as the Euler-Poinsot rigid body the Kepler problem and the stability of elliptic equilibria). However many efforts have made perturbation theory results suitable to such systems (7–9).
5. Analyticity is sufficient.