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.