One symmetry does not imply integrability

Abstract

We show that that Bakirov's counterexample (which had been checked by computeralgebra methods to order 53) to the conjecture that one nontrivial symmetry of an evolution equation implies innitely many is indeed a counterexample. To prove this we use the symbolic method of Gel'fand-Dikii and p-adic analysis. We also formulate a conjecture to the eect that almost all equations in the family considered by Bakirov have at most nitely many symmetries. This conjecture depends on the solution of a diophantine problem, which we explicitly state.