In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms

Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms?

@Deane: I took 'the Ricci flow' to mean the PDE system itself which one can think of as a submanifold of an appropriate jet bundle $J$ over $M\times\mathbb{R}$. Then a 'symmetry' would a self-diffeomorphism of $J$ that carries solutions to solutions (thought of via their $k$-jet graphs). @Carlo: You probably want to be more precise about what 'operations' you allow. Let $\mathcal{S}$ be the set of all Ricci-flat metrics on $\mathbb{R}^n$ and let $\sigma:\mathcal{S}\to\mathcal{S}$ be any mapping whatsoever. This is an 'operation that transforms one solution to another solution', no? –  Robert Bryant May 27 at 0:43