  We study complete, simply-connected manifolds with special holonomy that are
toric with respect to their multi-moment maps. We consider the cases where
there is a connected non-Abelian symmetry group containing the torus. For
$\mathrm{Spin}(7)$-manifolds, we show that the only possibility are structures
with a cohomogeneity-two action of $T^{3} \times \mathrm{SU}(2)$. We then
specialise the analysis to holonomy $G_{2}$, to Calabi-Yau geometries in real
dimension six and to hyperK\"ahler four-manifolds. Finally, we consider weakly
coherent triples on $\mathbb{R} \times \mathrm{SU}(2)$, and their extensions
over singular orbits, to give local examples in the $\mathrm{Spin}(7)$-case
that have singular orbits where the stabiliser is of rank one.
