Baecklund transformations and non abelian nonlinear evolution equations. A novel Baecklund chart

Third order Non Abelian Nonlinear Evolution Equations, such as potential Ko- rteweg deVries (pKdV), Korteweg deVries (KdV) as well as different versions of modified Korteweg deVries (mKdV), and the Korteweg deVries interacting soliton and Korteweg de- Vries singuarity manifold equations are here considered. In particular a new B ̈acklund Chart is constructed. Notably it, on one side, generalizes the Abelian B ̈acklund Chart presented in [24] and, on the other one, extends the noncommutative B ̈acklund Chart obtained in [12] to incorporate further non Abelian operator equations. The Ba ̈cklund transformations which link the various nonlinear evolution equations, allow to write the corresponding recursion operators and, hence, the corresponding whole hierarchies of nonlinear operators equations are connected.