  The Laplacian flow is a geometric flow introduced by Bryant as a way for
finding torsion free $G_2$-structures. If the flow is $S^1$-invariant then it
descends to a flow of $SU(3)$-structures on a $6$-manifold. In this article we
derive expressions for these evolution equations. In our search for examples we
discover the first inhomogeneous shrinking solitons, which are also gradient.
We also show that any compact non-torsion free soliton admits no infinitesimal
symmetry.
