On Testing Integrability

We demonstrate, using the symbolic method together with p-adic and resultant methods, the existence of systems with exactly one or two generalized symmetries. Since the existence of one or two symmetries is often taken as a sure sign (or as the definition) of integrability, that is, the existence of symmetries on infinitely many orders, this shows that such practice is devoid of any mathematical foundation. Extensive computations show that systems with one symmetry are rather common, and with two symmetries are fairly rare, at least within the class we have been considering in this paper.