The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem

Abstract

We review the Calogero-Fran\c{c}oise integrable system, which is a generalization of the Camassa-Holm system. We express solutions as (twisted) Higgs bundles, in the sense of Hitchin, over the projective line. We use this point of view to (a) establish a general answer to the question of linearization of isospectral flow and (b) demonstrate, in the case of two particles, the dynamical meaning of the theta divisor of the spectral curve in terms of mechanical collisions. Lastly, we outline the solution to the inverse problem for CF flows using Stieltjes' continued fractions.