Soliton solutions of the mean curvature flow and minimal hypersurfaces

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