Mink${}_3\times S^3$ solutions of type II supergravity

We initiate the classification of supersymmetric solutions of type II supergravity on ${\mathbb{R}}^{1,2}\times S^3\times M_4$. We find explicit local expressions for all backgrounds with either a single Killing spinor or two of equal norm, up to PDE’s. We show that the only type II AdS${}_4\times S^3$ solution is the known $\mathcal{N}=4$ AdS${}_4$ background obtained from the near-horizon limit of intersecting D2-D6 branes. Various known branes and intersecting brane systems are recovered, and we obtain a novel class of ${\mathbb{R}}^{1,2}\times S^2\times S^3$ solutions in IIA.