Tangency and a ruled surface associated with a Hitchin system

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 Definition 4. In particular, we will calculate the dimension of the moduli space of Hitchin covers satisfying the tangency condition to exhibit non-vacuousness of this definition. 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.