Perturbations of the Lagrange and Hess-Appelrot cases in the rigid body dynamics

The Lagrange case in the rigid body dynamics is completely integrable, with a family of invariant tori supporting periodic or quasi-periodic motion. We study perturbations of this case. In the non-periodic case the KAM theory predicts no changes in the evolution but in the periodic cases one expects existence of isolated invariant tori with some hyperbolic limit cycles, which are studied using Melnikov functions. We prove these phenomena in the case when the invariant torus is close to so-called critical circle. That approach is analogous to our previous analysis of the Hess-Appelrot case. In particular, we show that the number of created limit cycles is bounded.