  Analytical perturbations of the Euler top are considered. The perturbations
are based on the Poisson structure for such a dynamical system, in such a way
that the Casimir invariants of the system remain invariant for the perturbed
flow. By means of the Poincar\'{e}-Pontryagin theory, the existence of limit
cycles on the invariant Casimir surfaces for the perturbed system is
investigated up to first order of perturbation, providing sharp bounds for
their number. Examples are given.
