In the same way, $M5\perp M5\perp M5$ can be transformed to $M2\perp M2\perp M5\perp M5$ by adding a wave.

To go from an $S^3$ to an $S^2$ one adds a monopole. This is based on the fact that $S^3$ can be written as a $U(1)$ bundle over $S^2$ (Hopf fibration). The $U(1)$ field is precisely the monopole field, explaining why we need a monopole in order to go from a solution involving an $S^3$ to a solution involving an $S^2$. Reduction or $T$-duality along the “Hopf isometry” of $S^3$ yields $S^2$ or $S^2 \xx S^1$, respectively. Similar procedures were also described in the recent papers [@DLP; @ITY]. To illustrate the mechanism we consider again the $M2 \perp M5$ intersection and add a monopole. This means that we replace the overall transverse space by a (single-center) euclidean Taub-NUT space (see [@EGH] p.363). The solution is given by[@Tse1] ds\^2&=&H\_2\^[-]{}H\_5\^[-]{}(-dt\^2 + dx\_1\^2) +H\_2\^[-]{}H\_5\^(dx\_2\^2) +H\_2\^H\_5\^[-]{}(dx\_3\^2++dx\_6\^2) &+& H\_2\^H\_5\^ \[H\_M\^[-1]{}(d+ Q\_M d )\^2 + H\_M (dr\^2 + r\^2 d Ø\_2\^2)\]. where $\psi$ is a periodic coordinate with period $4 \pi Q_M$, $(\theta, \f)$ are coordinates on $S^2$ and H\_M = 1 + [Q\_M r]{} ,   H\_[2,5]{}=1+[Q\_[2,5]{}r]{}. For small $r$ one may neglect the one in the 