What is an indicator system?

D.P. Zhelobenko gave a realization of irreducible representaions of a simple Lie algebra as polynomial functions on the upper-triangular matrices that satisfy a system of PDE called the indicator system. This realization is very useful for investigating the branching of an irreducible representation under a restriction of the algebra. Using the procedure of restriction in the cases of $gl_n$ and $sp_{2n}$, Zhelobenko managed to construct a basis in an irreducible representation called the Gelfand-Tsetlin basis.
In the talk I will explain how one can solve the indicator system for different Lie algebras, how one can establish relations between solution spaces for different Lie algebras, and how using these relations one can construct a Gelfand-Tsetlin type basis.