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

Rigidity of SU(2,2|2)-symmetric solutions in Type IIB