  We give a complete description of the two-dimensional moduli spaces of stable
Higgs bundles of rank 2 over complex projective line 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.
