  We study the eleven dimensional supergravity equations which describe a low
energy approximation to string theories and are related to M-theory under the
AdS/CFT correspondence. These equations take the form of a non-linear
differential system, on $\mathbb{B}^7\times\mathbb{S}^4$ with the
characteristic degeneracy at the boundary of an edge system, associated to the
fibration with fiber $\mathbb{S}^4.$ We compute the indicial roots of the
linearized system from the Hodge decomposition of the 4-sphere following the
work of Kantor, then using the edge calculus and scattering theory we prove
that the moduli space of solutions, near the Freund--Rubin states, is
parametrized by three pairs of data on the bounding 6-sphere.
