This paper was motivated by the observation that after quickly finding a number of hierarchies (mKdV, Sawada-Kotera, Kaup-Kuperschmidt) soon after finding that KdV was integrable, nothing more was found for polynomial scalar evolution equations linear in the highest order derivative. In this paper we prove that under some mild conditions on the equations one can put a uniform bound on the order of the recursion operator of any such hierarchy. We do this using the symbolic method, introduced by… CONTINUE READING