Integrable systems whose spectral curve is the graph of a function

Abstract

For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C=\{(\lambda ,z)\mid z=A(\lambda )\}$ of a function $A(\lambda )$. Those integrable systems provide an interesting ``toy model'' of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which $A(\lambda )$ lives on a cylinder, a torus or a Riemann surface of higher genus.

latex2e, 15 pages, no figure; (v2) typos in eq. (25) etc. are corrected; (v3) a few typos are corrected. This article will be published in CRM Proceedings and Lecture Notes vol. 37