Abelian versus non-abelian baecklund charts:some remarks

Connections via B ̈acklund transformations among different non- linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via B ̈acklund transfor- mations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities be- tween Ba ̈cklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The B ̈acklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non- commutative cases, are richer than the corresponding commutative ones.