ON A THEOREM OF HALPHEN AND ITS APPLICATION TO INTEGRABLE SYSTEMS

Abstract

Abstract. We extend Halphen’stheorem which characterizesthe solutions of certain nth-order differential equationswith rational coefficientsand meromorphic fundamental systems to a first-order n×n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connectionswith completely integrable modelsof the Calogero-Moser-type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles. 1