  Motivated by the Extremal Vanishing Horizon (EVH) black holes, their near
horizon geometry and the EVH/CFT proposal, we construct and classify solutions
with (local) SO(2,2) symmetry to four and five dimensional
Einstein-Maxwell-Dilaton (EMD) theory with positive, zero or negative
cosmological constant Lambda, the EMD-$\Lambda$ theory, and also $U(1)^4$
gauged supergravity in four dimensions and $U(1)^3$ gauged supergravity in five
dimensions. In four dimensions the geometries are warped product of AdS3 with
an interval or a circle. In five dimensions the geometries are of the form of
warped product of AdS3 and a 2d surface $\Sigma_2$. For the
Einsten-Maxwell-$\Lambda$ theory we prove that $\Sigma_2$ should have a U(1)
isometry, a rigidity theorem in this class of solutions. We also construct all
d dimensional Einstein vacuum solutions with $SO(2,2) \times U(1)^{d-4}$ or
$SO(2,2) \times SO(d-3)$ isometry.
