One Symmetry Implies Symmetry-Integrability for Scalar Evolution Equations

We prove two general results on generalized symmetries for equations of the form ut = um + f(u, u1, . . . , um−1), where f is a formal (differential) power series starting with terms that are at least quadratic. The first result states that any higher order symmetry must be also a differential polynomial if f is a differential polynomial of order less than m − 1. The method is to estimate the orders of homogeneous components inductively using the Faa de Bruno formula. It is necessary to the second result which asserts that, without an extra condition, the existence of one nontrivial symmetry implies the existence of infinitely many (not only for polynomial equations). In addition, a structure theorem concerning homogeneous differential polynomials of fixed degree is obtained by the approach of symmetric function. MSC 2000: 37K10; 37K05; 35A30; 37L20