  Classes of third order non-Abelian evolution equations linked to that of
Korteweg-de Vries-type are investigated and their connections represented in a
non-commutative B\"acklund chart, generalizing results in [Fuchssteiner B.,
Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to
be hereditary, thereby allowing the results to be extended to hierarchies. The
present study is devoted to operator nonlinear evolution equations: general
results are presented. The implied applications referring to finite-dimensional
cases will be considered separately.
