I will present certain birational maps that are solutions of the parametric entwining Yang-Baxter equation. These maps are obtained via the refactorisation problem of certain Darboux transformations associated with the Lax operators of certain soliton PDEs. I will also present various dynamical properties of the derived maps, such as existence of invariants and associated symplectic or Poisson structures, and I will prove their complete integrability in the Liouville sense, where possible. Then I will describe the generalisation of such maps over Grassmann algebras using refactorisation of products of supermatrices, i.e. Darboux transformations with bosonic and fermionic entries. I will use the analogue of the characteristic polynomial, which in this non- commutative setting is the characteristic function, to define an analogue of a spectral curve. The latter can be used to obtain invariants of these maps involving Grassmann variables. New higher dimensional commutative maps can be obtained fixing the or- der of the Grassmann algebra Γ(n) and I will discuss integrability properties of these derived commutative maps.