Spectral curves for the rogue waves

Here we find the spectral curves, corresponding to the known rational or quasi-rational solutions of AKNS hierarchy equations, ultimately connected with the modeling of the rogue waves events in the optical waveguides and in hydrodynamics. We also determine spectral curves for the multi-phase trigonometric, hyperbolic and elliptic solutions for the same hierarchy. It seams that the nature of the related spectral curves was not sufficiently discussed in existing literature.