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

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds