There is developed a current algebra representation scheme for reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space and its application to quantum Hamiltonian and symmetry operators in case of quantum integrable spatially many-and one-dimensional dynamical systems. As examples, we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially many-and one-dimensional models as generalized oscillatory, Calogero-Sutherland, Coulomb type and nonlinear Schrödinger dynamical systems of spinless bose-particles.group Diff(R 3 ) of the space R 3 and the Schwarz space of smooth real valued functions on R 3 . Moreover, the corresponding quantum Hamiltonian operators of the Schrödinger type in the Hilbert space Φ, as it appeared to be very surprising, possess a very nice factorized structure, completely determined by this groundstate vector |Ω) ∈ Φ. This fact posed a very interesting and important problem of studying the related mathematical structure of these factorized operators and the correspondence to the classical Hamiltonian many-particle non-relativistic systems generating them, specified by some kinetic and interparticle potential energy.Analytical studies in modern mathematical physics are strongly based on the exactly solvable physical models which are of great help in understanding their mathematical and frequently hidden physical nature. Especially, the solvable models are of great importance in quantum many-particle physics, amongst which one can single out the oscillatory systems and Coulomb systems, modelling phenomena in plasma physics, the well known Calogero-Moser and Calogero-Moser-Sutherland models, describing a system of many particles on an axis, interacting pair-wise through long range potentials, modelling both some quantum-gravity and fractional statistics effects. In this work we developed investigations of local quantum current algebra symmetry representations in suitably renormalized representation Hilbert spaces, suggested and devised before by G.A. Goldin with his collaborators, having further applied their results to constructing the related factorized operator representations for secondly-quantized many-particle integrable Hamiltonian systems. The main technical ingredient of the current algebra symmetry representation approach consists in the weak equivalence of the initial many-particle quantum Hamiltonian operator to a suitably constructed quantum Hamiltonian operator in the factorized form, strictly depending only on its ground state vector. The latter makes it possible to reconstruct the initial quantum Hamiltonian operator in the case of its strong equivalence to the related factorized Hamiltonian operator form, thereby constructing, as a by-product, the corresponding N-particle groundstate vector for arbitrary N ∈ Z + . Being uniquely defined by means of the Bethe groundstate vector representation in the Hilbert space, the analyzed factorized operator structure of quantum compl...