Integrable Hamiltonian systems with two degrees of freedom associated with holomorphic functions

We focus on integrable systems with two degrees of freedom that are integrable in the Liouville sense and are obtained as real and imaginary parts of a polynomial (or entire) complex function in two complex variables. We propose definitions of the actions for such systems (which are not of the Arnol'd–Liouville type). We show how to compute the actions from a complex Hamilton–Jacobi equation and apply these techniques to several examples including those recently considered in relation to perturbations of the Ruijsenaars–Schneider system. These examples introduce the crucial problem of the semiclassical approach to the corresponding quantum systems.