Periodic orbits for vector fields with nondegenerate first integrals

We give a geometric proof in the particular case of nondegenerate first int egrals. of a theorem by Moser about the existence of periodic orbits on eac h level set of the integral, in a neighbourhood of a singular point of a ve ctor field satisfying a nonresonance hypothesis. We use the same geometric approach to deal with the resonance case, obtaining a generalization of pre vious results by Sweet. (C) 2001 Academic Press.