Gel'fand–Zakharevich systems and algebraic integrability: the Volterra lattice revisited

In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton–Jacobi (H–J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov–Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton–Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel'fand–Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H–J equations. We will then show how this setting works by framing results by Veselov and Penskoï about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables.