Separation of Variables for Hitchin Systems with the Structure Group $\mathrm {SO}(4)$ on Genus $2$ Curves

Sets of points that determine spectral curves can be regarded as phase coordinates of Hitchin systems. We address the problem of finding trajectories of Hitchin systems in these coordinates and solve it for systems with the structure groups $\mathrm {SO}(4)$ and $\mathrm {SL}(2)$ on genus $2$ curves. Our method consists in transferring the straight line windings from invariant tori, which are given by the Prymians of the spectral curves for Hitchin systems with simple classical structure groups. The transfer is carried out by means of an analog of the Jacobi inversion map, which does not exist for Prymians in general but can be defined in the two cases in question.