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Analytical perturbations of the Euler top are considered. The perturbationsare based on the Poisson structure for such a dynamical system, in such a waythat the Casimir invariants of the system remain invariant for the perturbedflow. By means of the Poincar\'{e}-Pontryagin theory, the existence of limitcycles on the invariant Casimir surfaces for the perturbed system isinvestigated up to first order of perturbation, providing sharp bounds fortheir number. Examples are given.

Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces