  We develop methods to study the singularities of certain $G_2$ cones related
to toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us to
determine the low energy gauge groups of chiral N=1 compactifications of
M-theory on a large family of such backgrounds, which includes the models
recently studied by Acharya and Witten. All M-theory compactifications
belonging to our family admit a $T^2$ of isometries, and therefore T-dual IIA
and IIB descriptions. We argue that reduction through such an isometry leads
generically to systems of weakly and strongly coupled IIA 6-branes, T-dual to
delocalized type IIB 5-branes. We find a simple criterion for the existence of
a `good' isometry which leads to IIA models containing only weakly-coupled
D6-branes, and construct examples of such backgrounds. Some of the methods we
develop may also apply to different situations, such as the study of certain
singularities in the hypermultiplet moduli space of N=2 supergravity in four
dimensions.
