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On a Theorem of Halphen and its Application to Integrable Systems

Fritz Gesztesy·Karl Unterkofler·Rudi Weikard
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Abstract

We extend Halphen's theorem which characterizes the solutions of certain $n$th-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order $n \times n$ 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.

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Fritz Gesztesy, Karl Unterkofler, Rudi Weikard. 2003-05-29. On a Theorem of Halphen and its Application to Integrable Systems. https://arxiv.org/abs/nlin/0305058

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