+\beta )} \left(e^{6 \alpha +2 \beta } (\cosh (2 \chi )-3)-2 \left(e^{4 \beta
   }+1\right) \cosh ^2(\chi )\right)\,,$$ is the superpotential determined by the supersymmetry variations, see Section \[BPSeqs\].

The potential, $\cP$, has three critical points [@Khavaev:1998fb] and those give rise to three $AdS_5$ vacua in this truncation:

-   The maximally supersymmetric critical point with $SO(6)$ global symmetry: $$\label{SO6fp}
    \alpha =\beta=\chi=0\;, \qquad \mathcal{P} = -\frac{3}{4}\;, \qquad L = \frac{2}{g}\;.$$

-   The $\mathcal{N}=2$ critical point with $SU(2) \times U(1)$ global symmetry: $$\label{PWfp}
    \alpha = \frac{1}{6}\log 2\;, \qquad \beta=0\;, \qquad \chi=\pm \frac{1}{2}\log 3\;, \qquad \mathcal{P} = -\frac{2^{4/3}}{3}\;, \qquad L = \frac{3}{2^{2/3}g}\;.$$

-   The non-supersymmetric $SU(3)$-invariant critical point: $$\label{SU3fp}
    \alpha =0\;, \qquad \beta=0\;, \qquad \chi= \frac{1}{2}\log (2\pm \sqrt{3})\;, \qquad \mathcal{P} = -\frac{27}{32}\;, \qquad L = \frac{2^{5/2}}{3g}\;.$$

The $AdS_5$ radius, $L$, is related to the critical value, $\cals P_*$, of the potential by $$\label{}
L^2\eql -{3\over g^2\,\cals P_*}\,.$$ Both supersymmetric points (i) and (ii) are also critical points of the superpotential and are perturbatively stable. The $SU(3)$-invariant critical point is stable within the truncation, but is perturbatively unstable in the full $\cN=8$ theory.

In the following we will concentrate on the supersymmetric $SU(2) \times U(1)$-invariant critical point (ii), from now on referred to as the 