The Laplacian flow is a geometric introduced by Bryant as way for finding torsion free $$G_2$$ -structures starting from closed one. If the invariant under $$S^1$$ action then it descends to of SU(3)-structures on 6-manifold. In this article we derive expressions these evolution equations. our search examples discover first inhomogeneous shrinking solitons, which are also gradient. We show that...