  We consider a completely integrable system of differential equations in
arbitrary dimensions whose phase space contains an open set foliated by
periodic orbits. This research analyzes the persistence and stability of the
periodic orbits under a nonlinear periodic perturbation. For this purpose, we
use the Melnikov method and Floquet theory to establish conditions for the
existence and stability of periodic orbits. Our approach considers periods of
the unperturbed orbits depending on the integrals and constant periods. In the
applications, we deal with both cases. Precisely, we study the existence of
periodic orbits in a perturbed generalized Euler system. In the degenerate
case, we analyze the existence and stability of periodic orbits for a perturbed
harmonic oscillator.
