Periodic Orbits for Vector Fields with Nondegenerate First Integrals

We give a geometric proof, in the particular case of nondegenerate first integrals, of a theorem by Moser about the existence of periodic orbits on each level set of the integral, in a neighbourhood of a singular point of a vector field satisfying a nonresonance hypothesis. We use the same geometric approach to deal with the resonance case, obtaining a generalization of previous results by Sweet