  The WZNW and string models are considered in the terms of the initial and
invariant chiral currents assuming that the internal and external torsions
coincide (anticoincide) and they are the structure constants of the
$SU(n),SO(n),$ $SP(n)$ Lie algebras. These models are the auxiliary problems in
order to construct integrable equations of hydrodynamic type. It was shown that
the WZNW and string models in terms of invariant chiral currents are integrable
for the constant torsion associated with the structure constants of the
$SU(2),$ $SO(3),$ $SP(2)$ and $SU(3)$ algebras only. The equation of motion for
the density of the first Casimir operator was obtained in the form of the
inviscid Burgers equation. The solution of this equation is presented through
the Lambert function. Also, a new equation of motion for the initial chiral
current was found.
  The integrable infinite hydrodynamic chains obtained from the WZNW and string
models are given in terms of invariant chiral currents with the $SU(2)$,
$SO(3)$, $SP(2)$ and with $SU(\infty)$, $SO(\infty)$, $SP(\infty)$ constant
torsions. Also, the equations of motion for the density of any Casimir operator
and new infinite dimensional equations of hydrodynamic type for the initial
chiral currents through the symmetric structure constant of $SU(\infty)$,
$SO(\infty),$ $SP(\infty)$ algebras are obtained.
