Tri-hamiltonian vector fields, spectral curves and separation coordinates

Since the mid-1990s the classical Hamiltonian theory of separation of variables has been actively developed via establishing connections with other approaches to integrability of Hamiltonian systems within both classical and quantum formalisms. Thus, the existence of bi-Hamiltonian and/or Lax structures has been shown to be very useful in the business of finding separable coordinates, which are employed to find a complete solution of the Hamilton-Jacobi equation for a given Hamiltonian system, thus leading to exact solutions of the latter. More specifically, in either case the associated geometric (i.e., the compatible Poisson bivectors of the bi-Hamiltonian approach) and algebro-geometric (i.e., the spectral curve of the Lax approach) structures serve as basic ingredients to construct separable coordinates. Lately the natural problem of establishing a link between separable coordinates obtained from the bi-Hamiltonian and Lax structures has received much attention. The authors consider a class of Hamiltonian systems for which two bi-Hamiltonian structures have been found (defined by a triplet of Poisson structures $(P_0, P_1, P_2)$). This setting, under some additional assumptions, makes it possible to construct a generalized set of separable coordinates corresponding to the separable coordinates obtained via the poles of the normalized Baker-Akhiezer function and the corresponding eigenvalues of the Lax operator in the Sklyanin approach based on the existence of the Lax structure.