Lax Pair Covariance and Integrability of Compatibility Condition

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 ZakharovShabat 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 .