On a Theorem of Halphen and its Application to Integrable Systems1

We extend Halphen’s theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and 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 connections with completely integrable models of 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.