Integrability of the Periodic Km System

1. The KM system In two seminal papers [9] for the modern theory of integrable systems Kac and vanMoerbeke introduced the system of o.d.e.’s u̇i = e ui+1 − ei−1 , i = 1, . . . , n, (1) where formally e0 = en+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 ui 7→ xi = ei is mapped to: ẋi = xi(xi+1 − xi−1) , i = 1, . . . , n, (2) (xn+1 = x0 = 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