  We construct many new invariant solutions to the Strominger system with
respect to a 2-parameter family of metric connections
$\nabla^{\varepsilon,\rho}$ in the anomaly cancellation equation. The ansatz
$\nabla^{\varepsilon,\rho}$ is a natural extension of the canonical 1-parameter
family of Hermitian connections found by Gauduchon, as one recovers the Chern
connection $\nabla^{c}$ for $({\varepsilon,\rho})=(0,\frac12)$, and the Bismut
connection $\nabla^{+}$ for $({\varepsilon,\rho})=(\frac12,0)$. In particular,
explicit invariant solutions to the Strominger system with respect to the Chern
connection, with non-flat instanton and positive $\alpha'$ 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-K\"ahler homogeneous spaces, obtained as the quotient by a lattice
of maximal rank of a nilpotent Lie group, the semisimple group
SL(2,$\mathbb{C}$) 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 $\nabla^{\varepsilon,\rho}$.
