ABSTRACT
The concept of the orbit in general was introduced by Kirilov and Kostant in the general study of Lie groups. Even field theoretic models in two dimension’s such as Weiss-Zumino-Witten models, can be constructed with the help of such co-adjoint orbit construction. The vectors in the carrier space of the co-adjoint representation are the linear forms on the Lie algebra. This chapter discusses an example of a dynamical system involving both commuting and anticommuting dependent variables using the AKS theorem. It shows how the moment map can be effectively used in conjunction with the AKS theorem to deduce and analyze new non-linear integrable problems. The chapter draws attention to the fact that it is also possible to use the concept of the co-adjoint orbit method even in the case of an algebra or groups whose members are not matrices.
