Soliton solutions of the mean curvature flow and minimal hypersurfaces

Let (M,g) be an oriented Riemannian manifold of dimension at least 3 and X a vector field on M. We show that the Monge-Amp\`ere differential system (M.A.S.) for X-pseudosoliton hypersurfaces on (M,g) is equivalent to the minimal hypersurface M.A.S. on (M,g') for some Riemannian metric g', if and only if X is the gradient of a function u, in which case g'=exp(-2u)g. Counterexamples to this equivalence for surfaces are also given.