x^{j} \, ,$$

where all the components are independent of $v$, where $v$ is defined by $V^{\mu} \partial_{\mu}\equiv \partial/\partial v$.

It can be checked that for all these metrics the KSE has solutions. These, then, are all the supersymmetric field configurations of $N=1,d=4$ SUGRA, but only those with $R_{\mu\nu}=0$ are supersymmetric solutions.

Example: $N=2,d=4$ Supergravity.
--------------------------------

With two Weyl spinors[^3] $\epsilon^{I}$ one can construct the following independent bilinears

-   A complex scalar $\bar{\epsilon}^{I}\epsilon^{J}\equiv
      M\varepsilon^{IJ}$

-   A Hermitean matrix of null vectors $V^{I}{}_{J\, \mu}\equiv
      i\bar{\epsilon}^{I} \gamma_{\mu}\epsilon_{J}$

The KSEs imply the following equations for the bilinears:

$$\begin{aligned}
\nabla_{\mu}M & \sim &  
F^{+}{}_{\mu\nu}V^{I}{}_{I}{}^{\nu}\, ,\label{eq:FV}\\
& & \nonumber \\
\nabla_{\mu}V^{I}{}_{J\, \nu} & \sim &
\delta^{I}{}_{J}[M F^{+}{}_{\mu\nu} 
+M^{*} F^{-}{}_{\mu\nu}]
-\Phi_{KJ\, (\mu}{}^{\rho}
\varepsilon^{KI}F^{-}{}_{\nu)\rho}
-\Phi^{IK}{}_{(\mu|}{}^{\rho}
 \varepsilon_{KJ} F^{+}{}_{|\nu)\rho}\ ,\\\end{aligned}$$

so the vector $V^{\mu}\equiv V^{I}{}_{I}{}^{\mu}$ is Killing and the other three are exact forms. The Fierz identities tell us that $V^{\mu}V_{\mu}\sim
|M|^{2}\geq 0$ can be timelike or null. When it is timelike, $V^{\mu}\partial_{\mu}\equiv \sqrt{2} \partial/\partial t$ and the metric can be put in the *conformastationary* form

$$\label{eq:conformastat}
ds^{2}  =  |M|^{2} (dt+\omega)^{2} -|M|^{-2} d\vec{x}^{\, 2}\, ,$$

where, for consistency, the 1-form $\omega$ has to be related to $M$ by

$$\label{eq:doeq}
d\omega= i|{M}|^{-2}{}^{\star} [{{M}  d{M}^{*}-{\rm c.c.}}]\, .$$

On the other hand, 