On the Integrability of Non-polynomial Scalar Evolution Equations

We show the existence of infinitely many symmetries for λ-homogeneous equations when λ=0. If the equation has one generalized symmetry, we prove that it has infinitely many and these can be produced by recursion operators. Identifying equations under homogeneous transformations, we find that the only integrable equations in this class are the Potential Burgers, Potential Modified Korteweg–de Vries, and Potential Kupershmidt Equations. We can draw some conclusions from these results for the case λ=−1 which, although theoretically incomplete, seem to cover the known integrable systems for this case.