  We analyze and completely describe the four cases when the Hitchin fibration
on a $2$-dimensional moduli space of irregular Higgs bundles over
$\mathbb{C}P^{1}$ 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 inelliptic fibrations having
this type of singular fiber at infinity.
