Closed $\mathrm{G}_2$-structures with $\mathbb{T}^3$-symmetry and hypersymplectic structures

We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure with $\mathbb{T}^3$-symmetry. Closed $\mathrm{G}_2$-structures $\varphi$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into two types according the orbits being non-isotropic or isotropic. Type I: if some orbit is non-isotropic, then the action is almost-free and $\varphi$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type II: if some orbit is isotropic, then the action is locally multi-Hamiltonian for $\varphi$. Moreover, the open and dense subset of principal orbits is foliated by $\mathbb{T}^3$-invariant hypersymplectic manifolds. If $\varphi$ is torsion-free, then for Type I, there arises another natural hypersymplectic structure, and a generalized Gibbons-Hawking Ansatz extending Madsen-Swann Ansatz is derived. For Type II, $\varphi$ is locally toric. Assuming moreover completeness and constant orbit volume, exactly three possibilities occur. Type Ia: orbits are purely non-isotropic non-associative, then the hypersymplectic 4-orbifold becomes a flat manifold. Type Ib: orbits are purely associative, then the $\mathbb{T}^3$-action is flat, and the hypersymplectic 4-orbifold becomes a hyperkähler 4-orbifold. Type II: orbits are isotropic, then all orbits are principal, and $\varphi$ is flat.