(e) 
\gamma^{a} +2\mathcal{E}^{\mu}(A)\} = 0\, .$$

On the other hand, the integrability conditions of the KSEs $\delta_{\epsilon}
\psi_{\mu} = \tilde{\cal D}_{\mu}\epsilon=0$ are

$$\begin{array}{l}
[\tilde{\cal D}_{\mu},\tilde{\cal D}_{\nu}]\epsilon
=
-{\textstyle\frac{1}{4}} 
\left\{
\left[
R_{\mu\nu}{}^{ab}
-e^{a}{}_{[\mu}T_{\nu]}{}^{b}
\right]\gamma_{ab}
+\nabla^{a} 
\left( F_{\mu\nu} 
+{}^{\star}F_{\mu\nu} \gamma_{5}\right)\gamma_{a}
\right\} \epsilon=0\, ,\\
\\
\\
\Rightarrow\,\,\,
\{ \mathcal{E}_{a}{}^{\mu} (e) \gamma^{a} 
+2[\mathcal{E}^{\mu}({A}) 
+\mathcal{B}^{\mu}(A) \gamma_{5}] \} 
\epsilon = 0\, .
\end{array}$$

In this case we get a more general formula from the integrability conditions, valid for the case in which the Bianchi identities are not satisfied. When they are satisfied we recover the KSIs, which is consistent since we have explicitly used the supersymmetry variations of the vector field in order to derive them, assuming, then, implicitly, that the Bianchi identities are satisfied.

The last formula (which we are also going to call KSI) has one important advantage over the original KSI: it is covariant under the $U(1)$ group of electric-magnetic duality rotations of the Maxwell and Bianchi identities that act as chiral rotations of the spinors.

Solving Tod’s problem
=====================

In 1983 showed in Ref. [@Tod:1983pm] that in $N=2,d=4$ SUGRA the problem could be completely solved using just integrability and consistency conditions. However, he used the Newman-Penrose formalism, unfamiliar to most particle physicists 