Vol. 40 (1997) REPORTS ON MATHEMATICAL PHYSICS No. 3 INTEGRABILITY OF THE PERIODIC KM SYSTEM ∗ Rui L. Fernandes Departamento de Matem´atica, Instituto Superior T´ ecnico, 1096 Lisboa Codex, Portugal (e-mail: rfern@math.ist.utl.pt) Jo˜ ao P. Santos Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA (e-mail: jsantos@math.stanford.edu) (Received March 3, 1997 ) We study the integrability of the periodic Kac-van Moerbeke system. We give a bi-hamiltonian formulation and a Lax pair containing a spectral parameter. Using Griffiths aproach we linearize the system on the Jacobian of the associated spectral curve. 1. The KM system In two seminal papers [9] for the modern theory of integrable systems Kac and van- Moerbeke introduced the system of o.d.e.’s ˙ u i = e ui+1 − e ui-1 , i =1,...,n, (1) where formally e u0 = e un+1 = 0. They showed, for example, that this system arises as a finite-dimensional aproximation of the famous KdV equations. Shortly after, Moser [10] showed that this system can be related to the classical Toda lattice, a rather well studied system. This meant also that interest shifted towards this latter system. Let us observe that system (1) under the change of variable u i → x i = e ui is mapped to: ˙ x i = x i (x i+1 − x i−1 ) , i =1,...,n, (2) (x n+1 = x 0 = 0). This is a Lotka-Volterra system, a classs of systems first studied by Volterra in his famous monograph [14]. There, Volterra introduced general systems of o.d.e’s of the form ˙ x j = ε j x j + 1 β j n  k=1 a jk x j x k , j =1,...,n, (3) * Supported in part by JNICT grant PBIC/C/MAT/2140/95 [1]