Hitchin fibrations on two-dimensional moduli spaces of irregular Higgs bundles with one singular fiber

We analyze and completely describe the four cases when the Hitchin fibration on a 2-dimensional moduli space of irregular Higgs bundles over the complex projective line has a single singular fiber. The case when the fiber at infinity is of type I_0^* is further analyzed, and we give constructions of all the possible configurations of singular curves in elliptic fibrations having this type of singular fiber at infinity.