$$G_2$$-structures on flat solvmanifolds

In this article we study the relation between flat solvmanifolds and $$G_2$$ -geometry. First, give a classification of 7-dimensional splittable using finite subgroups $$\mathsf{GL}(n,\mathbb {Z})$$ for $$n=5$$ $$n=6$$ . Then, look closed, coclosed divergence-free -structures compatible with metric on them. particular, provide explicit examples compact manifolds torsion-free -structure whose holonomy is cyclic contained in , admitting -structure.