regularity of the admissible solutions and we study in very close detail several examples in Section \[sec-n2d4bhs\]. Section \[sec-conclusion\] contains our conclusions.

Timelike BPS solutions of $N=2,d=4$ SUEGRA {#sec-d4BPS}
==========================================

It was recently shown in Ref. [@Meessen:2006tu] that all the supersymmetric solutions in the timelike class of $N=2,d=4$ supergravity coupled to $n$ vector multiplets[^6] can be constructed by setting the $2\bar{n}=2(n+1)$ components of a real, symplectic vector $\mathcal{I}=(\mathcal{I}^{\Lambda},\mathcal{I}_{\Lambda})$ equal to $2\bar{n}=2(n+1)$ real functions harmonic on 3-dimensional Euclidean space[^7]

$$\label{eq:stabilization}
\mathcal{I} 
\equiv
\left( 
  \begin{array}{c}
  \mathcal{I}^{\Lambda} \\  
  \mathcal{I}_{\Lambda} \\  
  \end{array}
\right)
\, ,
\hspace{1cm}
\partial_{m}\partial_{m} \mathcal{I}^{\Lambda}
=
\partial_{m}\partial_{m} \mathcal{I}_{\Lambda}=0\, ,
\hspace{1cm}
\Lambda=0,1,\cdots,n\, .$$

This real section $\mathcal{I}$ enters the theory as the imaginary part of the section $\mathcal{V}/X$, where $\mathcal{V}$ is the covariantly-holomorphic canonical section defining special geometry:

$$\label{eq:SGDefFund}
\mathcal{V} = 
\left(
\begin{array}{c}
\mathcal{L}^{\Lambda}\\
\mathcal{M}_{\Sigma}\\
\end{array}
\right) \;\; \rightarrow \;\;
\left\{
\begin{array}{lcl}
\langle \mathcal{V}\mid\mathcal{V}^{*}\rangle 
& \equiv & 
\mathcal{L}^{*\, \Lambda}\mathcal{M}_{\Lambda} 
-\mathcal{L}^{\Lambda}\mathcal{M}^{*}_{\Lambda}
= -i\, , \\
& & \\
\mathfrak{D}_{i^{*}}\mathcal{V} & = & (\partial_{i^{*}}-
{\textstyle\frac{1}{2}}\partial_{i^{*}}\mathcal{K})\mathcal{V} =0 \, ,\\
& & \\
\langle\mathfrak{D}_{i}\mathcal{V}\mid\mathcal{V}\rangle & = & 0 \, .
\end{array}
\right.$$

$X$ on the other hand is proportional to the complex, scalar bilinear constructed out of the Killing spinors: supersymmetry and consistency of the solutions 