Complete reduction of oscillators in resonance p : q

This paper extends to any type of resonance (p:q) the Lissajous transformat ion that handles the resonance (1:1) in a Hamiltonian composed of two harmo nic oscillators. The manifolds of constant energy for such a system an two- dimensional surfaces of revolution that are spheres for the resonance 1:1, spheres pinched once for the resonances (1:q) when 1<q, and spheres pinched twice for the resonances (p:q) when 1<p<q. The extended Lissajous transfor mation is valid for resonant pseudo-oscillators (a nondefinite quadratic fo rm), which allows us to find that the reduced phase flow lies on an unbound ed surface of revolution.