  We consider balanced metrics on complex manifolds with holomorphically
trivial canonical bundle, most commonly known as balanced
$\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian
geometry and string theory, since they provide the ideal setting for the
Hull-Strominger system. In this paper, we provide a non-existence result for
balanced non-K\"ahler $\rm{SU}(3)$-structures which are invariant under a
cohomogeneity one action on simply connected six-manifolds.
