Classical integrable systems and billiards related to generalized Jacobians

We study some classical integrable systems of dynamics (the Euler top in space, the asymptotic geodesic motion on an ellipsoid) which are linearized on unramified coverings of generalized Jacobian varieties. We find explicit expressions for so called root functions living on such coverings which enable us to solve the problems in terms of generalized theta-functions. In addition, general and asymptotic solutions for ellipsoidal billiards and the billiard in an ellipsoidal layer are obtained.