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We give the general form of supersymmetric backgrounds with two realsupercharges of M-theory and type IIA supergravity (with non-zero Romans massin general) of the form $\mathbb{R}^{1,d} \times \M_8$, d=1,2, oneight-dimensional manifolds with SU(4) structure. We point out a subtlety inthe integrability theorems for low-dimensional supersymmetriccompactifications. As a special case we examine Calabi-Yau flux vacua and weshow that unbroken supersymmetry does not in general require the four-form fluxto be (2,2) or primitive. Our results could be used to construct novelhigher-dimensional analogues of the Klebanov-Strassler geometry. In the case ofM-theory large-volume Calabi-Yau flux vacua our results are in agreement withpartial supersymmetry breaking in three-dimensional N=2 supergravity.Alternatively, the conditions for supersymmetry can be expressed in terms of areal `superpotential' in accordance with three-dimensional N=1 supergravity. Wepresent explicit examples of M-theory flux vacua on K3 \times K3, which howeverdo not appear to possess F-theory duals with four-dimensional Poincar\'einvariance.

IIA supergravity and M-theory on manifolds with SU(4) structure