A Super-Integrable Hierarchy and Its Super-Hamiltonian Structure

It has been an important and interesting topic in soliton theory for searching for new Lax integrable or Liouville integrable systems as many as possible such that they are associated with certain evolution equations with physical meaning. The simple and efficient method to obtain continuous or discrete integrable systems was proposed Tu. Ma developed it and called it Tu scheme. By taking advantage of it, a family of integrable systems[4−8] associated with physics backgrounds have been obtained. With the development of soliton theory, super-integrable systems and their super-Hamiltonian structures have been receiving growing attention.[9−15] In 1990, Hu proposed the super-trace identity firstly. Then the super-trace identity was mentioned and applied to establish the super-Hamiltonian structure of super-integrable systems. Recently, Professor Ma gave a systematic proof of super-trace identity and the expression of its constant γ. Meanwhile, he presented the super-Hamiltonian structures of super-AKNS hierarchy and super-Dirac Hierarchy for application.The main ideas in Ref.[12] are as follows: Let A be a commutative super algebra over R or C, and G be a matrix loop super algebra over A with the nondegenerate Killing form. Based on G, we consider the following isospectral problem { φx = U(u, λ)φ, φt = V (u, λ)φ, λt = 0, (1)