Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system

Abstract

This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the $\rmSU(2)$ Calogero-Gaudin system is used for a case study of this approach. A clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic Lax equation. The coordinates of these moving points give a new set of Darboux coordinates called the spectral Darboux coordinates. The system of isomonodromic deformations is thereby converted to a non-autonomous Hamiltonian system in the spectral Darboux coordinates. The Hamiltonians turn out to resemble those of a previously known isomonodromic system of a second order scalar ODE. The two isomonodromic systems are shown to be linked by a simple relation.