  We propose a new construction of compact non-K\"ahler Calabi-Yau manifolds
with balanced metrics and study the Strominger system on them. In particular,
we obtain explicit solutions to the Strominger system with degeneracies on
$\Sigma_g\times T^4$, where $\Sigma_g$ is an immersed minimal surface of genus
$g\geq3$ in $T^3$.
