SYMMETRIES OF THE KDV EQUATION AND 4 HIERARCHIES OF THE INTEGRODIFFERENTIAL KDV EQUATIONS

Using the inverse strong symmetry of the Korteweg-de Vries (KdV) equat ion on the trivial symmetry and tau0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are expressed expl icitly by the multi-integrations of the Jost function of the KdV equat ion and constitute an infinite dimensional Lie algebra together with t wo hierarchies of the known symmetries. Contrary to the general belief , the time-independent symmetry groups of the KdV and mKdV equations a re non-Abelian and the infinite dimensional Lie algebras of the KdV an d mKdV equations are nonisomorphic though two equations are related by the Miura transformation. Starting from these sets of symmetries, fou r hierarchies of the integrodifferential KdV equations, which can be s olved by the Schrodinger inverse scattering transformation method, are obtained. Some of these hierarchies enjoy a common strong symmetry an d/or same local conserved densities.