$M$.

-   The KSIs guarantee that these three different sets of equations (plus the equations of the scalar fields, if any) are complicated combinations a a reduced number of simple equations involving a reduced number of scalar unknowns. Solving these equations for the scalar unknowns gives full solutions of the theory. The tricky part is, usually, identifying the right variables that satisfy simple equations and finding these equations as combinations of the Maxwell, Einstein etc. equations.

-   Finally, with the results obtained, the KSEs have to be solved, which may lead to additional conditions on the fields.

Let us see how this recipe works in the examples considered before.

Example: $N=1,d=4$ Supergravity.
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With one ([Majorana]{}) [Killing spinor $\epsilon$]{} the only bilinear that one can construct is a real vector bilinear $V_{\mu}$ which is always null. $V_{\mu}$ is also covariantly constant (i.e. it is a Killing vector and $V_{\mu}dx^{\mu}$ is an exact 1-form, which allows us to write $V_{\mu}dx^{\mu}=du$):

$$\delta_{{\epsilon}}{\psi_{\mu}}=
\nabla_{\mu}{\epsilon}=0\, ,\,\,\, \Rightarrow   
\nabla_{\mu} {V_{\nu}}=0\, ,\,\,\, 
{R^{\mu}{}_{\nu}}{V^{\nu}}=0\,
,\,\,\,\, ({\bar{\epsilon}}
{R^{\mu}{}_{a}}\gamma^{a}{\epsilon}=0)\, .$$

All the metrics with covariantly constant null vectors are Brinkmann pp-waves and have the form

$$ds^{2} = 2 du (dv + K du +A_{\underline{i}}  d x^{i})
+\tilde{g}_{\underline{i}\underline{j}}  dx^{i} d 