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.