Symmetric and asymmetric separation of variables for an integrable case of the complex Kirchhoff's problem

Abstract

In the present paper we develop a method of the vector fields Zi[9] in the theory of separation of variables. For an integrable case of the complex Kirchhoff's problem on e⁎(3), which has been never considered before, we construct—with the help of this method—two types of separation of variables (SoV): symmetric and asymmetric ones. Our asymmetric SoV is unusual: it is characterized by the quadratures containing differentials defined on two different curves of separation. It is a direct analogue of asymmetric SoV for the Clebsch model [17]. In the case of symmetric SoV both curves of separation are the same. This case has an additional bonus: on zero level set of one of the Casimir functions it yields a direct analogue of the famous Weber-Neumann separated coordinates. They are also considered in the present paper in some details.