Smooth prime integrals for quasi-integrable Hamiltonian systems

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