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We analyze and completely describe the four cases when the Hitchin fibrationon a $2$-dimensional moduli space of irregular Higgs bundles over$\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber atinfinity is of type $I_0^*$ is further analyzed, and we give constructions ofall the possible configurations of singular curves inelliptic fibrations havingthis type of singular fiber at infinity.

Hitchin Fibrations on Two-Dimensional Moduli Spaces of Irregular Higgs Bundles with One Singular Fiber