and suited only for $d=4$. Thus, there were no further results until 1995, when Tod, using again the same methods, solved partially the problem in $N=4,d=4$ SUGRA [@Tod:1995jf]. Then, in 2002, Gauntlett, Gutowski, Hull, Pakis and Reall proposed to translate the Killing spinor equation to tensor language and they solved the problem in minimal $N=1,d=5$ SUGRA [@Gauntlett:2002nw]. This opened the gates to new results: in 2002 the problem was solved in *gauged* minimal $N=1,d=5$ SUGRA [@Gauntlett:2003fk], in 2003 in minimal $N=(1,0),d=6$ SUGRA [@Gutowski:2003rg; @Chamseddine:2003yy] and gauged $N=2,d=4$ SUGRA [@Caldarelli:2003pb], and in 2004 and 2005 in gauged minimal $N=1,d=5$ SUGRA coupled to Abelian vector multiplets [@Gutowski:2004yv; @Gutowski:2005id] and in $N=4,d=4$ SUGRA [@Bellorin:2005zc], completing the work started by Tod on this theory.

There is by now a well-defined recipe to attack this problem (at least in low dimensions) starting with only one assumption: the existence of one Killing spinor $\epsilon$. The recipe consists in the following steps:

-   Translate the [Killing spinor]{} equations and KSIs into tensorial equations.

    With the Killing spinor $\epsilon$ one can construct scalar, vector, and $p$- form bilinears $M\sim \bar{\epsilon} \epsilon\, ,\hspace{.5cm}
      V_{\mu}\sim \bar{\epsilon} \gamma_{\mu} \epsilon\, , \cdots$ that are related by Fierz identities. These bilinears satisfy certain 