  The general approach to chain equations derivation for the function generated
by a Miura transformation analog is developing to account evolution (second Lax
equation) and illustrated for
  Sturm-Liouville differential and difference operators. Polynomial
differential operators case is investigated. Covariant sets of potentials are
introduced by a periodic chain closure. The symmetry of the system of equation
with respect to permutations of the potentials is used for the direct
construction of solutions of the chain equations.
  A "time" evolution associated with some Lax pair is incorporated in the
approach via closed t-chains. Both chains are combined in equations of a
hydrodynamic type. The approach is next developed to general Zakharov-Shabat
differential and difference equations, the example of 2x2 matrix case and NS
equation is traced.
