We construct many new invariant solutions to the Strominger system with respect to a 2-parameter family of metric connections ∇ε,ρ in the anomaly cancellation equation. The ansatz ∇ε,ρ is a natural extension of the canonical 1-parameter family of Hermitian connections found by Gauduchon, as one recovers the Chern connection ∇c for (ε,ρ)=(0,12), and the Bismut connection ∇+ for (ε,ρ)=(12,0). In particular, explicit invariant solutions to the Strominger system with respect to the Chern connection, with non-flat instanton and positive α′ are obtained. Furthermore, we give invariant solutions to the heterotic equations of motion with respect to the Bismut connection. Our solutions live on three different compact non-Kähler homogeneous spaces, obtained as the quotient by a lattice of maximal rank of a nilpotent Lie group, the semisimple group SL(2,C) and a solvable Lie group. To our knowledge, these are the only known invariant solutions to the heterotic equations of motion, and we conjecture that there is no other such homogeneous space admitting an invariant solution to the heterotic equations of motion with respect to a connection in the ansatz ∇ε,ρ

The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the (restricted) holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian, then any invariant balanced J-Hermitian structure provides solutions of the Strominger system.