  We will generalize the Treibich-Verdier theory about elliptic solitons to a
Hitchin system by constructing a particular ruled surface and we will propose a
generalization of a tangency condition associated with elliptic solitons to a
Hitchin system. In particular, we will calculate the dimension of the moduli
space of Hitchin covers satisfying the tangency condition. With this new point
of view, we will see a subtle relation between the characterizations of
coverings and the singularities of divisors in a particular algebraic surface.
