  A class of n-dimensional Poisson systems reducible to an unperturbed harmonic
oscillator shall be considered. In such case, perturbations leaving invariant a
given symplectic leaf shall be investigated. Our purpose will be to analyze the
bifurcation phenomena of periodic orbits as a result of these perturbations in
the period annulus associated to the unperturbed harmonic oscillator. This is
accomplished via the averaging theory up to an arbitrary order in the
perturbation parameter e. In that theory we shall also use both branching
theory and singularity theory of smooth maps to analyze the bifurcation
phenomena at points where the implicit function theorem is not applicable. When
the perturbation is given by a polynomial family, the associated Melnikov
functions are polynomial and tools of computational algebra based on Gr\"obner
basis are employed in order to reduce the generators of some polynomial ideals
needed to analyze the bifurcation problem. When the most general perturbation
of the harmonic oscillator by a quadratic perturbation field is considered, the
complete bifurcation diagram (except at a high codimension subset) in the
parameter space is obtained. Examples are given.
