Abstract: When considering compactifications of heterotic string theory, the Hull--Strominger system arises. It is a system of non-linear PDEs involving an invariant nowhere-vanishing holomorphic (3,0)-form and a metric which is conformally balanced. In this talk, we will describe the Hull--Strominger system and then investigate the existence of the geometric structures required by it in the cohomogeneity one setting. We provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. This talk is based on a joint work with F. Salvatore.