  We consider a certain ${\cal N}=1$ supersymmetric, $SO(3)\times SO(3)$
invariant, subsector of the $\omega$-deformed family of $SO(8)$-gauged ${\cal
N}=8$ four-dimensional supergravities. The theory contains two scalar fields
and two pseudoscalar fields. We look for stationary points of the scalar
potential, corresponding to AdS vacua in the theory. One of these, which breaks
all supersymmetries but is nonetheless stable, is new. It exists only when
$\omega\ne 0$. We construct supersymmetric domain wall solutions in the
truncated theory, and we give a detailed analysis of their holographic dual
interpretations using the AdS/CFT correspondence. Domain walls where the
pseudoscalars vanish were studied previously, but those with non-vanishing
pseudoscalars, which we analyse numerically, are new. The pseudoscalars are
associated with supersymmetric mass deformations in the CFT duals. When
$\omega$ is zero, the solutions can be lifted to M-theory, where they approach
the Coulomb-branch flows of dielectric M5-branes wrapped on $S^3$ in the deep
IR.
