  Reconsidering the M2-brane solutions of d=11 supergravity with a transverse
Englert flux introduced by one of us in 2016, we present a new purely group
theoretical algorithm to solve Englert equation based on a specific embedding
of the PSL(2,7) group into $\mathrm{Weyl[\mathfrak{e}_7]}$. The aforementioned
embedding is singled out by the identification of $\mathrm{PSL(2,7)}$ with the
automorphism group of the Fano plane. Relying on the revealed intrinsic
PSL(2,7) symmetry of Englert equation and on the new algorithm we present an
exhaustive classification of Englert fluxes. The residual supersymmetries of
the corresponding M2-brane solutions associated with the first of the 8 classes
into which we have partitioned Englert fluxes are exhaustively analyzed and we
show that all residual d=3 supersymmetries with $\mathcal{N} \in
\left\{1,2,3,4,5,6\right\}$ are available. Our constructions correspond to a
particular case in the category of M2-brane solutions with transverse self-dual
fluxes.
