0 \ . \label{O26} \end{equation} This solution seems to be problematic close to $r=0$. Indeed, if we compute the Ricci scalar we obtain $R=0$, nevertheless, $R_{\mu\nu}R^{\mu\nu}\sim \frac{3}{32 L^4 r^4}+\ldots . $ close to $r=0$. The solution is singular and we will not study it further. It is interesting to notice that a family of \emph{non-SUSY} fixed point solutions exists. Indeed, we can consider the situation where $B,U,V$ and $A'(r)=a_1$ are constant. For $S^2$ and $H_2$, we find then that the Einstein equations impose $U=V=0$ and \begin{equation} 8+e^{-4B}z^2-4 \ a_1^2=0 \,, ~~~~~~ 4-z^2 e^{-4B}+\kappa \ e^{-2B}=0. \label{O12} \end{equation} Where $\kappa=+1$ for $S^2$ and $\kappa=-1$ for $H_2$. The solution is, \begin{equation} z^2= e^{2B} \left( 4 e^{2B} + \kappa \right) \,, ~~~~~~ a_1^2=3 + \kappa \ \frac{e^{-2B}}{4} \label{O13} \end{equation} For the $S^2$ case the range of parameters is, \begin{equation} B \in \mathbb{R} \,, ~~~~~~ z \in \mathbb{R}-\{0\} \,, ~~~~~~ a_1 \in \left( -\infty,\sqrt{3}\right) \cup \left( \sqrt{3}, +\infty \right) \,, \label{O14} \end{equation} while for ${H_2}$ we find, \begin{equation} B \in \Big[-\ln 2, +\infty \Big) \,, ~~~~~~ z \in \mathbb{R} \,, ~~~~~~ a_1 \in \Big( -\sqrt{3},-\sqrt{2} \ \Big] \cup \Big[ \ \sqrt{2}, \sqrt{3} \ \Big) \ . \label{O15} \end{equation} Notice that in the case of $H_2$, there is a non-SUSY solution with $z=0$, hence no fibration between the hyperbolic plane and the Reeb vector $\eta$. The SUSY fixed point in eq.~\eqref{O20} is part of the family in eq.~\eqref{O15}, with $z=-\frac{1}{3}$. For the $T^2$ we also find an $AdS_3$ solution, \begin{equation} a_1^2=3 \,, ~~~~~~ z ^2=4e^{4B} \,, ~~~~~~ U=0 \,, ~~~~~~ V=0 \ . \label{T7} \end{equation} Notice that $z \in \mathbb{R}-\{0\} $. It would be interesting to study the stability of these solutions. A possible way to do this is to write these solutions in the language of a five or a three dimensional gauged supergravity, like the ones in~\cite{Gauntlett-ml-2009zw,Gauntlett-ml-2009zw.m001,Gauntlett-ml-2009zw.m002,Gauntlett-ml-2009zw.m003}. Once in that formalism, to study the fluctuations of the different scalar fields (that determine the sizes of the internal space and the $H_2$). It is likely that these solutions admit small fluctuations whose mass is below the BF bound, hence making the solution unestable. We leave this point to be studied in the future. This completes our presentation of what we will refer as `twisted' solutions. By twisted we mean solutions where a gauge field is switched on the Riemann surface generating a fibration between $\Sigma_2$ and the R-symmetry direction. In the following, we will present a background that also contains an $AdS_3$ factor, it flows in the UV to an $AdS_5$, but the field content and the mechanism of SUSY preservation are different from the ones above. ]]>