  We continue to study Lax (L-A, U-V) pairs (LP) joint covariance with respect
to Darboux transformations (DT) as a classification principle. The scheme is
based on a comparison of general expressions for the transformed coefficients
of LP and its Frechet derivative. We use the compact expressions of the DT via
some version of nonabelian Bell polynomials. It is shown that one more version
of Bell polynomials, so-called binary ones, form a convenient basis for the
invariant subspaces specification. Some non-autonomous generalization of KdV
and Boussinesq equations are discussed in the context.
  Trying to pick up restrictions at minimal operator level we consider
Zakharov-Shabat - like problem . The subclasses that allow a DT symmetry
(covariance at the LP level) are considered from a point of view of dressing
chain equations. The case of classic DT and binary combinations of elementary
DT are considered with possible reduction constraints of Mikhailov type
(generated by an automorphism). Examples of Liuville - von Neumann equation for
density matrix are referred as illustrations .
