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

B\"acklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel B\"acklund Chart