Minkowski${}_4$ $\times$ $S^2$ solutions of IIB supergravity

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.