  We classify $\mathcal N = 2$ Minkowski$_4$ solutions of IIB supergravity with
an $SU(2)_R$ symmetry geometrically realized by an $S^2$-foliation in the
remaining six dimensions. For the various cases of the classification, we
reduce the supersymmetric system of equations to PDEs. These cases often
accommodate systems of intersecting branes and half-maximally supersymmetric
AdS$_{5,6,7}$ solutions when they exist. As an example, we analyze the AdS$_6$
case in more detail, reducing the supersymmetry equations to a single
cylindrical Laplace equation. We also recover an already known linear dilaton
background dual to the $(1,1)$ Little String Theory (LST) living on NS5-branes,
and we find a new Minkowski$_5$ linear dilaton solution from brane
intersections. Finally, we also discuss some simple Minkowski$_4$ solutions
based on compact conformal Calabi-Yau manifolds.
