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 $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $T^3\times \mathrm{S}\mathrm{U}(2)$. We then specialise the analysis to holonomy $G_2$, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $\mathbb{R}\times \mathrm{S}\mathrm{U}(2)$, and their extensions over singular orbits, to give local examples in the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-case that have singular orbits where the stabiliser is of rank one.