Non-Kaehler Heterotic String Compactifications with non-zero fluxes and constant dilaton

We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold $M_3$ satisfy the heterotic equations of motion.