  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.
