Bäcklund transformations, well known to represent a key tool in the study of nonlinear evolution equations, are shown to allow the construction of a net of nonlinear links, termed Bäcklund chart, connecting Abelian as well as non-Abelian equations. In particular, Bäcklund transformations are applied to reveal algebraic properties enjoyed by nonlinear evolution equations they connect. The present study concerns third-order nonlinear evolution equations, termed KdV-type, which are all connected to the KdV equation. The Abelian wide Bäcklund chart connecting these nonlinear evolution equations is recalled. Then, the links, originally established in the case of Abelian equations, are shown to conserve their validity when non-Abelian counterparts are considered. In addition, the non-commutative case reveals a richer structure since there may be more than a single non-Abelian counterpart of the same Abelian equation. Reduction from the non-commutative to the commutative case allows to show the connection of the KdV equation with KdV eigenfunction equation, in the scalar case. The main result presented refers to the KdV eigenfunction equation: some explicit solutions it admits are constructed on application of an invariance property proved via Bäcklund transformations. These, to the best of the authors? knowledge, new solutions represent an example of the powerfulness of the method devised. Matrix solutions of the mKdV equations, recently obtained, are mentioned in the closing remarks to stress the powerfulness of the method.