After reading a bit more about Perelman's entropy and gradient solitons, I came up with a hunch, which I must test. Non-singular solitons can be regarded as critical points of Perelman's entropy, or fixed points of the Ricci flow characterized by $$R_{ij} + \lambda g_{ij} + \nabla _i \nabla _j f = 0 .$$ My question is about the function $f$ when it does exist. Is there any flexibility in the choice of $f$ for a particular soliton? That is to say, at any fixed time, is $f$ always uniquely determined? Please, a couple or more of explicit solitons with the accompanying $f$'s would be highly appreciated so as to run some tests with the F- and W- functionals.