Ren-integrable and ren-symmetric integrable systems

A new type of symmetry, ren-symmetry, describing anyon physics and corresponding topological physics, is proposed. Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics, super-symmetric gravity, super-symmetric string theory, super-symmetric integrable systems and so on. Super-symmetry and Grassmann numbers are, in some sense, dual conceptions, and it turns out that these conceptions coincide for the ren situation, that is, a similar conception of ren-number (R-number) is devised for ren-symmetry. In particular, some basic results of the R-number and ren-symmetry are exposed which allow one to derive, in principle, some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems. Training examples of ren-integrable KdV-type systems and ren-symmetric KdV equations are explicitly given.