  We complete the classification of Mink$_4$ solutions preserving
$\mathcal{N}=2$ supersymmetry and SU(2) R-symmetry parameterised by a round
$S^2$ factor. We consider eleven-dimensional supergravity and relax the
assumptions of earlier works in type II theories. We show that, using chains of
dualities, all solutions of this type can be generated from one of two master
classes: an SU(2)-structure in M-theory and a conformal Calabi-Yau in type IIB.
Finally, using our results, we recover AdS$_5\times S^2$ solutions in M-theory
and construct a compact Minkowski solution with Atiyah-Hitchin singularity.
