We summarize the foliation approach to ${cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $mathrm{AdS}_3$ spaces for the case when the internal part $xi$ of the supersymmetry generator is chiral on some proper subset ${cal W}$ of $M$. In this case, a topological no-go theorem implies that the complement $Msetminus {cal W}$ must be a dense open subset, while $M$ admits a singular foliation ${bar {cal F}}$ (in the sense of Haefliger) which is defined by a closed one-form $boldsymbol{omega}$ and is endowed with a longitudinal $G_2$ structure. The geometry of this foliation is determined by the supersymmetry conditions. We also describe the topology of ${bar {cal F}}$ in the case when $boldsymbol{omega}$ is a Morse form.
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