  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.
