Lie-algebraic approach to nonlocal symmetries of integrable systems

A recursion operator is derived for a large class of equations that can be integrated by the inverse scattering method. For the obtained hierarchies of integrable equations a method is proposed for constructing an algebra of nonlocal symmetries. The complete set of dynamical variables corresponding to them is found. It is shown that all the nonlocal variables are the integrals of the densities of conservation laws. The structure of the obtained systems is illustrated by the example of the Zakharov–Shabat (AKNS) hierarchy.