Asymmetric separation of variables for the extended Clebsch and Manakov models

Abstract

In the present paper we develop a method of the vector fields Zi[8] of the bi-hamiltonian theory of separation of variables. Using a modification of this method we construct non-Stäckel asymmetric variable separation for three-dimensional extension of the Clebsch model, which is bi-hamiltonially equivalent to the system of interacting Manakov (Shottky-Frahm) and Euler tops. Our SoV is unusual: it is characterized by two curves of separation of different genus. We explicitly construct coordinates and momenta of separation, the corresponding reconstruction formulae and the Abel-type quadratures for the considered three-dimensional extension of the Clebsch and Manakov tops.