Symmetric and asymmetric separated variables. II. A second integrable case of the complex Kirchhoff model

Abstract

In the present paper, we find a second new integrable case of the complex Kirchhoff model on e*(3). We show that the obtained integrable model possesses two types of separation of variables (SoV): symmetric and asymmetric ones. The asymmetric SoV—similar to that of the Clebsch model (F. Magri and T. Skrypnyk, arXiv:1512.04872) and of the first integrable case of the complex Kirchhoff model (T. Skrypnyk, J. Geom. Phys. 172, 104418 (2022)]—is characterized by two different curves of separation. In the case of symmetric SoV, both curves of separation are the same. This case—similar to the integrable case of the complex Kirchhoff model of Skrypnyk, J. Geom. Phys. 172, 104418 (2022)—yields a direct analog of the famous Weber–Neumann separated coordinates. The obtained separation curves and Abel-type equations for the constructed model can be viewed as trigonometric degenerations of the corresponding separation curves and Abel-type equations of the Clebsch model [F. Magri and T. Skrypnyk, arXiv:1512.04872 and T. Skrypnyk, J. Geom. Phys. 135, 204–218 (2019)], while separation curves and Abel-type equations of Skrypnyk, J. Geom. Phys. 172, 104418 (2022) can be viewed as their rational degenerations.