Separation of variables for Hitchin systems with the structure group $SO(4)$, on genus two curves

Sets of points giving spectral curves can be regarded as phase coordinates of Hitchin systems. We address the problem of finding out trajectories of Hitchin systems in those coordinates. The problem is being solved for the systems with structure groups $SO(4)$ and $SL(2)$ on genus 2 curves. Our method is a transfer of straight line windings of fibers of the Hitchin map, which are given by Prymians of the spectral curves for the 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.