Separation of variables for the Clebsch model. New cases

Abstract:
We will make a brief review of two modern approaches to the separation of variables for the integrable systems on the dual spaces to the Lie algebras, namely of the Lax-pair based “magic recipe” and of the bi-hamiltonian approach or “method of the vector fields Z”. Using the last approach we will find two sets of separated (in the sense of the Hamilton-Jacobi) variables for the Clebsch model. One of them generalizes the famous Neumann separated variables from the special onto the general coadjoint orbit of E(3). Another is completely new even in the (sub)case of Neumann.