A tri-Hamiltonian route to spectral curves.

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P_0,P_1,P_2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P_1-λP_0) and (P_2-μP_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.