  We give the general form of supersymmetric backgrounds with two real
supercharges of M-theory and type IIA supergravity (with non-zero Romans mass
in general) of the form $\mathbb{R}^{1,d} \times \M_8$, d=1,2, on
eight-dimensional manifolds with SU(4) structure. We point out a subtlety in
the integrability theorems for low-dimensional supersymmetric
compactifications. As a special case we examine Calabi-Yau flux vacua and we
show that unbroken supersymmetry does not in general require the four-form flux
to be (2,2) or primitive. Our results could be used to construct novel
higher-dimensional analogues of the Klebanov-Strassler geometry. In the case of
M-theory large-volume Calabi-Yau flux vacua our results are in agreement with
partial supersymmetry breaking in three-dimensional N=2 supergravity.
Alternatively, the conditions for supersymmetry can be expressed in terms of a
real `superpotential' in accordance with three-dimensional N=1 supergravity. We
present explicit examples of M-theory flux vacua on K3 \times K3, which however
do not appear to possess F-theory duals with four-dimensional Poincar\'e
invariance.
