B\"acklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations.

The KdV eigenfunction equation is considered: some explicit solutions are constructed. These, to the best of the authors' knowledge, new solutions represent an example of the powerfulness of the method devised. Specifically, B\"acklund transformation are applied to reveal algebraic properties enjoyed by nonlinear evolution equations they connect. Indeed, B\"acklund 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 "B\"acklund chart", connecting Abelian as well as non Abelian equations. The present study concerns third order nonlinear evolution equations which are all connected to the KdV equation. In particular, the Abelian wide B\"acklund 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 related to the multiplicity of non-Abelian equations which correspond to the same Abelian one. Reduction from the nc to the commutative case allow to show the connection of the KdV equation with KdV eigenfunction equation, in the "scalar" case. Finally, recently obtained matrix solutions of the mKdV equations are recalled.