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.