ALGEBRAIC STRUCTURE OF THE GRADIENT-HOLONOMIC ALGORITHM FOR LAX INTEGRABLE NONLINEAR DYNAMICAL-SYSTEMS .2. THE REDUCTION VIA DIRAC AND CANONICAL QUANTIZATION PROCEDURE

The generalized theory of the R-structure on affine operator Lie algebras is used to construct a complete theory of Lax integrable nonlinear dynamical systems in multidimensions. The operator bi-Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator Korteweg-de Vries and Benney-Kaup dynamical systems. As an important by-product of the developed algebraic theory, the Dirac canonical quantization problem is solved almost completely for the Neumann-Bogoliubov-type oscillatory dynamical system on spheres' associated via Moser with the spectral moment map on an affine associative metrized Lie coalgebra with a one-parameter gauge two-cocycle. Some remarks are given on the problem of extending the developed algebraic theory to the case of Lax integrable dynamical systems on discrete manifolds.