A model of nonlinear oscillations is developed. The model decouples at once and nonlinear normal modes result. The solutions describe nonisochronous motion on invariant tori in phase space. Among the orbits there is a dense set on which the normal frequencies are in resonance. This oscillating system serves to illustrate many of the ideas of nonlinear dynamics, particularly the concept of integrability. The integrable system also provides the point of departure for an exploration of Hamiltonian chaos.