Eq. (\[eq:FV\]) gives

$$F^{+}  \sim  |M|^{-2} \{V\wedge dM +i {}^{\star}[V\wedge dM] \}\, .$$

The KSIs are satisfied if Eq. (\[eq:doeq\]) is satisfied. It can be seen that, then, any metric and 2-form field strength of the above form admit Killing spinors. On the other hand, all the equations of motion are combinations of the simple equation in 3-dimensional Euclidean space

$$\vec{\nabla}^{\,2}{M}^{-1}=0\, .$$

Thus, solving this equation for some $M$ gives us a supersymmetric solution of all the equations of motion (all the fields are determined by $M$). These solutions of the Einstein-Maxwell theory are the Israel-Wilson-Perjés family [@kn:IW; @kn:Pe].

The case in which $V$ is null is very similar to the $N=1$ case and we will not study it here in detail for lack of space.

Tod’s problem in $N=4,d=4$ supergravity
=======================================

This theory can be obtained by toroidal compactification on $T^{6}$ of $N=1,d=10$ SUGRA [@Chamseddine:1980cp] (the effective field theory of the Heterotic String) and subsequent (consistent) truncation of the matter vector fields. The 10- and 4-dimensional fields are related as indicated in Fig. \[fig:fields\].

$
 \begin{psmatrix}[rowsep=1.5,colsep=0.2]
 d=10, N=1
 &
 [name=e10]{\{e^{a}{}_{\mu},}
 &
 [name=B10]{B_{\mu\nu},}
 &
 [name=d10]{\phi,}
 &
 [name=p10]{\psi_{\mu},}
 &
 [name=c10]{\chi\}}
 &
 [name=V10]{\{V^{R}{}_{\mu},}
 & 
 [name=pm10]{\psi^{R}\}}
 & 
 &
 \\
 d=4, N=4 
 &
 [name=e4]{\{e^{a}{}_{\mu},}
 