three-dimensional anti-de Sitter spacetime[^6]. This is exactly the same near-horizon geometry as that of the $M2\perp M5$ intersection without a wave. Adding the wave takes us to a different form of the $adS_3$ metric (provided that no global identifications are made). To see this let us make the coordinate transformation[^7] x=x’-t’, t=(x’+t’), e\^[2]{} + 1 = r’\^2. This brings (\[adS3W\]) to the form \[BTZ\] ds\^2 = - [(r’\^2 -1)\^2 r’\^2]{} dt’\^2 + [r’\^2 (r’\^2 -1)\^2]{} dr’\^2 +r’\^2 (dx’-[1 r’\^2]{}dt’)\^2 This is the standard form of a massive extremal BTZ black hole [@BTZ] with mass and angular momentum $M=J=2$ in a space of cosmological constant $\L=-1/l^2$ equal to $-1$. These values simply reflect the fact that we have set all charges equal to $1$ in (\[adS3W\]). Notice, however, that the coordinate $x'$ in (\[BTZ\]) is not periodic. It is well-known that the BTZ black hole is locally isometric to $adS_3$, i.e. there is a coordinate transformation that brings (\[BTZ\]) to the anti-de Sitter metric. For non-extremal BTZ black holes this transformation is easy to write down. In the extremal case the situation is more complicated and one needs to consider an infinite number of patches that cover the black hole spacetime 