Abstract. A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.

Abstract. A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation is constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed by means of the gradient-holonomic and geometric methods.

We establish some connections between Lie rings, their derivation rings and generalized derivations rings.