are going to rederive these results using our notation and to relate them to the KSIs. We also want to improve the previous derivations by making explicit use of the knowledge of all the supersymmetric configurations.

Let us consider single, static, asymptotically flat, spherically symmetric, black-hole-type solutions of $N=2,d=4$ supergravity coupled to vector multiplets: they are given by real harmonic functions of the form

$$\label{eq:Iforspherical}
\mathcal{I} = \mathcal{I}_{\infty} +\frac{q}{r}\, ,$$

which is the general choice compatible with the assumptions. The metric can be conveniently written in spherical coordinates as

$$ds^{2} = 2|X|^{2}dt^{2} -\frac{1}{2|X|^{2}}[dr^{2}+r^{2}d\Omega^{2}_{(2)}]\, .$$

This metric describes black holes if

$$-g_{rr}=\frac{1}{2|X|^{2}}\stackrel{r\rightarrow \infty}{\longrightarrow} 
1+\frac{2M}{r}\, ,$$

is always finite for finite $r$, whence $M$, which is the mass, must be positive. Further, we have to require

$$\frac{1}{2|X|^{2}} \stackrel{r\rightarrow 0}{\longrightarrow} 
\frac{A}{4\pi r^{2}}>0\, ,$$

which imposes the existence of an event horizon with area $A> 0$ at $r=0$ instead of a naked singularity.

The existence of attractors (fixed points) of the scalar fields follows from the fact that in supersymmetric configurations, the scalars satisfy first-order differential equations, as follows immediately from the Killing spinor equations associated to the gaugino supersymmetry transformation rule:

$$\label{eq:gaugsusyrule}  
\delta_{\epsilon}\lambda^{Ii}  =  
i\not\!\partial Z^{i}\epsilon^{I} 
+\epsilon^{IJ}\not\!G^{i\, +}\epsilon_{J}=0\, .$$

To derive the needed 