Symmetries and integrable systems

Symmetry plays key roles in modern physics especially in the study of integrable systems because of the existence of infinitely many local and nonlocal generalized symmetries. In addition to the fundamental role to find exact group invariant solutions via Lie point symmetries, some important new developments on symmetries and conservation laws are reviewed. The recursion operator method is important to find infinitely many local and nonlocal symmetries of (1 + 1)-dimensional integrable systems. In this paper, it is pointed out that a recursion operator may be obtained from one key symmetry, say, a residual symmetry. For (2 + 1)-dimensional integrable systems, the master-symmetry approach and the formal series symmetry method are reviewed. For the discrete systems, the symmetry related discrete KP hierarchy and the BKP hierarchy are also discussed. One believes that all the solutions of integrable models may be obtained by means of symmetry approach because the Darboux transformations and algebro-geometric solutions can be obtained from the localization of nonlocal symmetries and the symmetry constraint approach. The conservation laws are used to find higher dimensional integrable system from lower dimensional ones via a deformation algorithm. The ren variable, an extension of the Grassmann variable, are introduced to find novel aspect on integrable theory. The super-integrable theory and super-symmetric integrable theory are extended to ren integrable and ren-symmetric integrable theories.