Symmetric Separation of Variables for the Extended Clebsch and Manakov Models

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Skrypnyk, Taras
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/214098
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Symmetric Separation of Variables for the Extended Clebsch and Manakov Models. Taras Skrypnyk. SIGMA 21 (2025), 066, 19 pages

In the present paper, using a modification of the method of vector fields ᵢ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for the 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 used to construct the 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.