  We provide a criterion in order to decide the stability of non-degenerate
equilibrium states of completely integrable systems. More precisely, given a
Hamilton-Poisson realization of a completely integrable system generated by a
smooth $n-$ dimensional vector field, $X$, and a non-degenerate regular (in the
Poisson sense) equilibrium state, $\overline{x}_e$, we define a scalar
quantity, $\mathcal{I}_{X}(\overline{x}_e)$, whose sign determines the
stability of the equilibrium. Moreover, if $\mathcal{I}_{X}(\overline{x}_e)>0$,
then around $\overline{x}_e$ there exist one-parameter families of periodic
orbits shrinking to $\{\overline{x}_e \}$, whose periods approach
$2\pi/\sqrt{\mathcal{I}_{X}(\overline{x}_e)}$ as the parameter goes to zero.
The theoretical results are illustrated in the case of the Rikitake dynamical
system.
