(\[eq:ksi2-2\]), we get the condition

$$\label{eq:VJm}
\langle\, \mathcal{V}/X \mid \, \mathcal{J}^{\underline{m}} \, \rangle =0\, .$$

Let us now consider the time component of the imaginary part of the KSI Eq. (\[eq:ksi2\]), Eq. (\[eq:ksi2-2\]):

$$\label{eq:IJt}
\langle\, \mathcal{I} \mid \, \mathcal{J}^{t} \, \rangle= 0\, .$$

To find the physical meaning of this condition we use the explicit form of the symplectic vector of vector field strengths $F$ for timelike BPS solutions Eq. (\[eq:esas\]):

$$\label{eq:jotamu}
\mathcal{J}^{\mu}=
\mathcal{E}^{\mu}=
 -({}^{\star}dF)^{\mu}= |X|^{2}\, 
(\partial_{\underline{m}}\partial_{\underline{m}}\mathcal{I})\,  V^{\mu}=
\frac{\delta^{\mu}{}_{t}}{\sqrt{2}}
\frac{\partial_{\underline{m}}\partial_{\underline{m}}\mathcal{I}}{\sqrt{|g|}}\, .$$

This result tells us that the KSIs Eq. (\[eq:VJm\]) are always satisfied and that the KSI Eq. (\[eq:IJt\]) is equivalent to the condition

$$\label{eq:iddi}
\langle\, \mathcal{I} \mid \, 
\partial_{\underline{m}}\partial_{\underline{m}}\mathcal{I} \, \rangle
=
\Im{\rm m}\, (\partial_{\underline{m}}\mathcal{W}_{\underline{m}})
= 0\, ,$$

which is nothing but the integrability condition for the equation determining $\omega$, which now has to be satisfied everywhere as a consequence of demanding unbroken supersymmetry *everywhere*. For the point-like sources considered above, these equations take the form

$$\sum_{A}\langle\, \mathcal{I} \mid \, q_{A} \, \rangle 
\delta^{(3)}(\vec{x}-\vec{x}^{A})/ \sqrt{|g|}=0\, .$$

The consequences of imposing this condition were first studied by Denef and Bates in Refs. [@Denef:2000nb; @Bates:2003vx] in the context of general $N=2,d=4$ supergravity, but was studied earlier by Hartle and Hawking in 