The heterotic G${}_2$-system on 2-step nilmanifolds endowed with principal torus bundles

We study the heterotic G${}_2$-system on 7-dimensional 2-step nilmanifolds $M=\mathrm{\Gamma }\mathrm{\backslash }N$ endowed with principal torus bundles. We first prove that every invariant G${}_2$-structure solving the system must be coclosed (under an additional calibration assumption when the dimension of the derived Lie algebra of $N$ is $3$). Then, we discuss the existence of solutions for all possible isomorphism classes of 7-dimensional 2-step nilpotent Lie algebras, and we provide examples with constant dilaton function both when the cosmological constant of the spacetime is zero and when it is nonzero.