Covariant Forms of Lax One-fieldoperators: from Abelian to Non-commutative

Polynomials in differentiation operators are considered. Joint covariance with respect to Darboux transformations of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechèt differential (first term of the Taylor series on prolonged space) to establish the operator forms. In the commutative (Abelian) case, as it was recently proved for the KP-KdV Lax operators, it results in binary Bell (Faa de Bruno) differential polynomials having natural bilinear (Hirota) representation. Now next example of generalized Boussinesq equation with variable coefficients is studied, the dressing chain equations for the pair are derived. For a pair of generalized Zakharov-Shabat problems a set of integrable (non-commutative) potentials and hence nonlinear equations are constructed altogether with explicit dressing formulas. Some non-Abelian special functions are introduced.