Mink3 × S3 solutions of type II supergravity

We initiate the classification of supersymmetric solutions of type II supergravity on R1,2×S3×M4. 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×4S3 solution is the known N=4 AdS4 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 R1,2×S2×S3 solutions in IIA.