Covariant forms of Lax one-field operators: from Abelian to non-commutative

Links of factorization theory, supersymmetry and Darboux transformations as isospectral deformations are considered in the context of quantum theory. The infinite chain equations for factorizing operators for a spectral problem are derived. A closure of the chain defines a symmetry of the system. Examples of matrix-differential operators of Pauli and Dirac type are analyzed.