Internal symmetries of integrable dynamical systems

Abstract

In the preceding two chapters we have illustrated the effectiveness of the method for integration of nonlinear dynamical systems based on the “spectral” properties of the operators entering representation (3.1.1) and taking values in a ℤ-graded Lie algebra. However, as we have mentioned it is impossible to describe all such systems since there is no uniform parametrization of structure constants of a generic Lie algebra and nor is there a general description of all their gradings. Therefore, though we can describe explicitly via the general construction the group element uniquely determining the corresponding solutions we cannot describe in terms of a Lie group or its Lie algebra a compact form of the equations themselves.