Symmetric Separation of Variables for the Extended Clebsch and Manakov Models

TL;DR

This paper develops a symmetric separation of variables method for extended Clebsch and Manakov models, revealing new symmetric coordinates and Abel-type equations, and highlighting differences from previous asymmetric approaches.

Contribution

It introduces a symmetric non-Stäckel separation of variables for extended integrable models, using a modified vector field method, and explicitly constructs separation coordinates and equations.

Findings

All separation curves have genus five.

Symmetric and asymmetric cases differ in vector field form.

Explicit separation coordinates and Abel equations are provided.

Abstract

In the present paper, using a modification of the method of vector fields $Z_i$ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-St\"ackel variable separation for three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields $Z$ used to construct separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.