On effective solution to the Hitchin systems

The integrable systems this talk is devoted to are invented by N.Hitchin in 1987, and bear his name. Since then they have been studied mainly from geometrical point of view while methods of solving them are much less developed. Originally, Hitchin systems have been defined in quite involved algebraic-geometric context but it has been revealed recently that they can be given an elementary definition by means of classical method of Separation of Variables (SoV). In the talk, we shall present this elementary definition and derive the fundamental fact that the trajectories of Hitchin systems are straight line windings of certain Abelian tori (Jacobians or Prymians of spectral curves, depending on the structure group of the system). The last is achieved by finding out the special Darboux coordinates where the system has a canonical form. Then we set the problem of expressing the trajectories in the original coordinates. To this end, we must reverse the above coordinate transform. In the case that the structure group is GL(n) we obtain an explicit theta function expression for the trajectories on this way. However for simple structure groups (for SO(n) for instance) we meet an obstruction related with peculiarities of the inversion problem on Prymians. We will argue that in principle there should be no obstruction for a numerical solving the system, may be except for typical difficulties like being of the inverse transform close to degenerate. Solution to this numerical problem would enable one to obtain a phase portrait of a Hitchin system that never has been done.