  Analytical perturbations of a family of finite-dimensional Poisson systems
are considered. It is shown that the family is analytically orbitally conjugate
in $U \subset \mathbb{R}^n$ to a planar harmonic oscillator defined on the
symplectic leaves. As a consequence, the perturbed vector field can be
transformed in the domain $U$ to the Lagrange standard form. On the latter, use
can be made of averaging theory up to second order to study the existence,
number and bifurcation phenomena of periodic orbits. Examples are given ranging
from harmonic oscillators with a potential and Duffing oscillators, to a kind
of zero-Hopf singularity analytic normal form.
