  We present a systematic search for Type IIB supergravity solutions whose
spacetimes include AdS5 and S2 factors, which would be candidate duals to N = 2
four-dimensional Superconformal field theories. The candidate solutions encode
the SU(2) R-symmetry geometrically on the S2 and an additional Killing vector
generates the U(1) R-symmetry. By analysing the Killing spinor equations we
show that no such solutions exist. This suggests that, if Type IIB backgrounds
dual to N = 2 SCFTs exist, the SU(2) R-symmetry is realised non-geometrically.
Finally, we also show that, in the context of both N = 1 and N = 2 Type IIB
backgrounds with an AdS5 factor, the only candidate U(1) R-symmetry Killing
vector directions are the ones that appear for generic values of the Killing
spinors; no further Killing vectors exist for special values of the Killing
spinors.
