Two-dimensional moduli spaces of rank 2 Higgs bundles over CP1 with one irregular singular point

We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over CP1 with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.