ABSTRACT: The Clebsch model is one of the few classical examples of the
dynamics of rigid bodies in the liquid where the equations of motion are
integrable in the sense of Liouville. The explicit solution of the
problem of the Hamilton-Jacobi separation of variables for this model
is, however, particularly hard and has remained unsolved for more than a
century. We have managed to solve this problem in several different
ways. In this talk we will present two variable separations for the
Clebsch model - symmetric and asymmetric ones. The asymmetric variable
separation is very unusual: it is characterized by the quadratures
containing differentials defined on two different curves of separation.
In the case of symmetric SoV both curves of separation are the same.
This case has a bonus: on a zero level set of one of the Casimir
functions it yields the famous Weber-Neumann separated coordinates. We
also find the explicit reconstruction formulae for the both sets of the
constructed separated variables and explicitly write the corresponding
Abel-type equations, completely resolving in such a way the
long-standing problem of variable separation for the Clebsch model.