On Euler Case in Rigid Body Dynamics and Jacobi Problem

Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth.