  We investigate the existence of half-BPS solutions in Type IIB supergravity
which are invariant under the superalgebra SU(2,2|2) realized on either AdS_5 x
S^2 x S^1 or AdS_5 x S^3 warped over a Riemann surface \Sigma with boundary. We
prove that, in both cases, the only solution is AdS_5 x S^5 itself. We argue
that this result provides evidence for the non-existence of fully back-reacted
intersecting D3/D7 branes with either AdS_5 x S^2 x S^1 x \Sigma or AdS_5 x S^3
x \Sigma near-horizon limits.
