On the existence of balanced metrics on 6-manifolds of cohomogeneity one

We consider complex manifolds with holomorphically trivial canonical bundle endowed with a balanced metric. In the compact case, such manifolds are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Strominger system. Let $M$ be a compact simply connected balanced cohomogeneity one 6-manifold endowed with an invariant nowhere-vanishing holomorphic $(3,0)$-form. We classify all possible principal parts of such manifolds, up to $G$-equivariant diffeomorphisms.