On a Theorem of Halphen and its Application to Integrable Systems

Details On a Theorem of Halphen and its Application to Integrable Systems On a Theorem of Halphen and its Application to Integrable Systems

2003-05-29

arxiv.org/abs/nlin/0305058v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTeX, 19 pages

Scientific paper

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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