Multi-toric geometries with larger compact symmetry

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 Spin (7) -manifolds, we show that the only possibility are structures with a cohomogeneity-two action of T^3 SU (2). We then specialise the analysis to holonomy G₂, to Calabi-Yau geometries in real dimension six and to hyperK\"ahler four-manifolds. Finally, we consider weakly coherent triples on R SU (2), and their extensions over singular orbits, to give local examples in the Spin (7) -case that have singular orbits where the stabiliser is of rank one.