The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with torsion to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat K$\ddot{\text{a}}$hler metrics coupled with Hermitian Yang-Mills equation on non-K$\ddot{\text{a}}$hler Calabi-Yau 3-folds. The Anomaly flow is a parabolic approach to understand the Hull-Strominger system initiated by Phong-Picard-Zhang. We show that in the setting of generalized Calabi-Gray manifolds, the Hull-Strominger system and the Anomaly flow reduce to interesting elliptic and parabolic equations on Riemann surfaces. By solving these equations, we obtain solutions to the Hull-Strominger system on a class of compact non-K$\ddot{\text{a}}$hler Calabi-Yau 3-folds with infinitely many topological types and sets of Hodge numbers. This talk is based on joint work with Zhijie Huang and Sebastien Picard.