not allowed, and in particular this means that no localized sources of angular momentum are allowed.

Any singular contributions to $\mathcal{J}^{0m}$ and $\mathcal{J}^{mn}$ must originate in the $R^{0m}$ components of the Ricci tensor; more precisely, they come from the term $\partial_{\underline{m}}(d\omega)_{\underline{m}\underline{n}}$, where $\omega$ is the 1-form that appears off-diagonally in the metric of the timelike supersymmetric solutions of $N=2,d=4$ supergravity Eq. (\[eq:metricd4\]). Therefore, using Eq. (\[eq:oidi\]) and defining the complex 3-dimensional vector $\vec{\mathcal{W}}$

$$\vec{\mathcal{W}}=
(\mathcal{W}_{\underline{m}})\equiv 
(\langle\, \mathcal{V}/X \mid \, 
\partial_{\underline{m}}\mathcal{I} \, \rangle)\, ,
\hspace{1cm}
\Im{\rm m}\, (\mathcal{W}_{\underline{m}})=
{\textstyle\frac{1}{4}}
\epsilon_{mnp}
(d\omega)_{\underline{n}\underline{p}}
=\langle\, \mathcal{I} \mid \, 
\partial_{\underline{m}}\mathcal{I} \, \rangle\, ,$$

we can translate the above KSIs, Eqs. (\[eq:J0mJmn\]), to the condition

$$\Im{\rm m}\, (\vec{\nabla}\times \vec{\mathcal{W}})=0\, ,$$

which has to be imposed everywhere. Actually, only the singular parts of this equation have to be taken into account since, dealing with solutions, the finite parts must be canceled in the equations of motion by other finite contributions. Therefore, from now on we will ignore all finite contributions to this equation.

Let us consider the real and imaginary parts of Eq. (\[eq:ksi2\]), namely Eq. (\[eq:ksi2-1\]) and (\[eq:ksi2-2\]). The real part gives us two important pieces of information: first, it tells us that the component $\mathcal{J}^{00}$ 