The symbolic method and cosymmetry integrability of evolution equations

We determine the existence of cosymmetries for the scalar evolution equations ut = uk + f(u, · · · , uk−1) by using the symbolic method and emphasize the role played by the permutation group in answering divisibility questions. In some special cases, we conclude that there is only one possible cosymmetry. The method can also be used to classify the evolution equations by generating functions. As an example, we give the complete list for 7th–order KdV-like equations with nontrivial conservation laws.