$adS_2$ one needs a $U(1)$ gauge field to support the $adS_2$ part. Putting a wave and reducing one obtains a $D0$ brane that precisely provides the required gauge field. The wave/monopole rule also allows for a determination of the killing spinors of all solutions once the killing spinors of the solutions containing only $adS_3$ and $S^3$’s are given.

We have organized the paper as follows. In the next section we present the wave/monopole rule. Using this rule we re-examine and organize the solutions with near-horizon geometry $adS_k \xx S^l \xx E^m$ studied in [@BPS1]. Then we present the new solutions. These contain two sphere factors. This seems to be the maximum number since one can only fit three 3-dimensional subspaces in $10$ dimensions (recall that the CFT of $adS_2$ and $S^2$ comes in terms of the CFT of $adS_3$ and $S^3$, respectively). In section 3, we analyze the supersymmetry enhancement of these configurations near the horizon. In particular, we explicitly work out the case of $adS_3 \xx S^3 \xx S^3 \xx E^2$. We use these results in section 4 to argue that string theory on the background given by the new solutions is dual to certain superconformal theories. In section 