Potential nonclassical symmetries and solutions of fast diffusion equation

The fast diffusion equation ut=(u−1ux)xut=(u−1ux)x is investigated from the symmetry point of view in development of the paper by Gandarias [M.L. Gandarias, Phys. Lett. A 286 (2001) 153]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.