results are summarized in Table 3.

One may wonder whether one may use the shift transformation to reach these solutions. In order to perform the shift transformation one needs to dualize each brane to a wave. For this to be possible each brane should be wrapped on a torus. However, the sphere parts of the solutions come from the worldvolume part of certain branes. So, if we want to end up with a solution that contains spheres we should not wrap the brane on a torus. Spheres and also $adS$ spaces have non-abelian isometries. So, it may still be possible to use non-abelian dualities to obtain an appropriate version of the shift transformation.

There is a simple rule (which we shall call the wave/monopole rule) that leads to solutions with $adS_2$ and $S^2$’s once a solution with $adS_3$ and $S^3$’s is given. To get an $adS_2$ one adds a wave to $adS_3$, whereas to get an $S^2$ one adds a monopole to $S^3$. One may anticipate these rules by looking at the corresponding CFTs. To get an $S^2$ one views $S^3$ as a Hopf fibration over $S^2$. The monopole solution precisely supplies the needed $U(1)$ monopole gauge field. Similarly, for 