  A non-abelian generalisation of a birational representation of affine Weyl
groups and their application to the discrete dynamical systems is presented. By
using this generalisation, non-commutative analogs for the discrete systems of
$A_n^{(1)}$, $n \geq 2$ type and of $d$-Painlev\'e equations with an additive
dynamic were derived. A coalescence cascade of the later is also discussed.
