has been considered by the following metric $$\label{warped bh metric}
ds^2\,=\,e^{-\frac{5}{3}\chi} ds^2_M +R^2 e^\chi
\left[\,\frac{e^{-4\eta}}{9}\bigg( d\psi+\sum_{i=1}^2 \cos\phi_i\bigg)^2
+\frac{e^{\eta}}{6} \sum_{i=1}^2 \Big(d\theta_i^2+\sin^2\theta_i\,d\phi^2_i\Big)
\,\right]$$ where the non compact space $M$ is given by $$ds^2_M \,=\,-\,g e^{-w} dt^2+\frac{dr^2}{g}+\frac{r^2}{R^2} \sum_{i=1}^3 dx_i^2\;.$$ The functions $\chi$, $\eta$, $g$ and $w$ depend on the coordinate $r$ only. The geometry (\[warped bh metric\]) is required to provide $AdS_5 \times T^{1,1}$ on the boundary, i.e. at large $r$.\
In [@Herzog:2009gd] the equations of motion coming from the effective Lagrangian have been solved numerically; nevertheless analytic formulae have been found in some limits. We are interested in the $T=0$ regime, for which the first term of a series expansion near the horizon is given. The novel feature is that the near horizon region is a warped product $AdS_2 \times \mathbb{R}^3 \times T^{1,1}$. As discussed in [@Herzog:2009gd], one can employ the symmetries of the problem to set to one both the $AdS$ radius and the position of the horizon, but we prefer to keep $r_0$ generic for clearness.\
The metric (\[warped bh metric\]) falls into the general class considered in the section \[more general ansatz section\] through the ansatz (\[ds2 general ansatz\]) by choosing $D=9$, $d=3$ and $$A(r)^2\,=\,g(r)
\hspace{2cm}
B(r)^2\,=\,\frac{r^2}{R^2}\;.$$ The analytic 