equations because they are made out of Killing spinors, for instance, if the KSE is of the general form

    $$\label{eq:Killingeq}
    \delta_{\epsilon} \psi_{\mu} =
    \tilde{\cal D}_{\mu}\epsilon=
    [\nabla_{\mu}+\Omega_{\mu}]
    \epsilon=0\, ,\,\,\,
    \Rightarrow\,\,
    \nabla_{\mu} M +2 \Omega_{\mu}M=0\, ,$$

    The set of all such equations for the bilinears should be equivalent to the original spinorial equation or at least it should contain most of the information contained in it (but, certainly, not all of it).

-   One of the vector bilinears (say $V_{\mu}$) is always a Killing vector which can be timelike or null. These two cases are treated separately.

-   One can get an expression of all the gauge field strengths of the theory using the Killing equation for those scalar bilinears: $\Omega_{\mu}$ is usually of the form $F_{\mu\nu}V^{\nu}$ and, then Eq. (\[eq:Killingeq\]) tells us that $F_{\mu\nu}V^{\nu}\sim \nabla_{\mu}
      \log M$. When $V$ is timelike this determines completely $F$ and, when it is null, it determines the general form of $F$. Of course, Eq. (\[eq:Killingeq\]) is an oversimplified KSE and in real-life situations there are additional scalar factors, $SU(N)$ indices etc.

-   The Maxwell and Einstein equations and Bianchi identities are imposed on those field strengths $F$, getting second order equations for the scalar bilinears 