Integrable WZNW model and string model of WZNW model type with $SU(n)$,$SO(n),SP(n)$ constant torsion and infinite hydrodynamic chains from $n$ to $\infty $

The integrability of WZNW model and string model of WZNW model type with constant $SU(2)$ torsion is investigated. The closed boson string model in the background gravity and antisymmetric B-field is considered as integrable system in terms of initial chiral currents. The model is considered under assumption that internal torsion related with metric of Riemann–Cartan space and external torsion related with antisymmetric B-field are (anti)coincide. New equation of motion and exact solution this equation was obtained for string model with constant $SU(2)$ torsion. New equations of motion and new Poisson brackets(PB) for infinite dimensional hydrodynamic chains was obtained for string model with constant $SU(n),SO(n),SP(n)$ torsion from $n$ to $\infty $.