  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.
