On $G_2$ manifolds with cohomogeneity two symmetry

We consider $G_2$ manifolds with a cohomogeneity two $\mathbb{T}^2\times \mathrm{SU}(2)$ symmetry group. We give a local characterization of these manifolds and we describe the geometry, including regularity and singularity analysis, of cohomogeneity one calibrated submanifolds in them. We apply these results to the manifolds recently constructed by Foscolo--Haskins--Nordstr\"om and to the Bryant--Salamon manifold of topology $S^3\times \mathbb{R}^4$. In particular, we describe new large families of complete $\mathbb{T}^2$-invariant associative submanifolds in them.