Smooth prime integrals in quasi integrable systems

A Hamiltonian with N degrees of freedom, analytic perturbation of a canonically integrable strictly nonisochronous analytic Hamiltonian, is considered. We show the existence of N functions on phase space and of class C^∞ which are prime integrals for the perturbed motions on a suitable region whose Lebesgue measure tends to fill locally the phase space as the perturbation’s magnitude approaches zero. An application to the perturbations of isochronous nonresonant linear oscillators is given.