We construct birational maps that satisfy the parametric set-theoretical Yang-Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable Nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense. [1,5,13,24,25]. Moreover, there exist solutions which can be associated to matrix refactorisation problems and can be further related to Darboux and Bäcklund transformations [15] of soliton equations.Due to the significance of equation (1), its connection with various fields of mathematical and physics and its numerous applications in the theory of Integrable Systems, it is important to construct new solutions and study their properties. The motivation to study solutions of the Yang-Baxter equation related to Darboux transformations arises naturally from some recently obtained results [15,23] where Yang-Baxter maps were obtained using Darboux transformations related to several integrable models. In particular, in [15], Darboux transformations associated with the nonlinear Schrödinger equation (NLS), derivative NLS (DNLS) and deformation of the DNLS (dDNLS) equation were employed in order to construct birational integrable Yang-Baxter maps, while in [23] a Yang-Baxter map on the sphere constructed in relation to a vectorial generalisation of the sine-Gordon equation. The Lax operators of the DNL and the dDNLS equation are invariant under the action of reduction groups isomorphic to Z 2 and D 2 , with rational dependence in the spectral parameter and having poles in C = C ∪ {∞} at generic and degenerate (nongeneric) orbits; in [15], only the operators corresponding to degenerate orbits were used. In the first part of this paper, we study the refactorisation problem of the product of two Darboux matrices associated to the DNLS Lax operator, invariant under the Z 2 reduction group with poles in C that form a single generic orbit under the action of the reduction group. In the second part, we employ different Darboux transformations related to the NLS operator and the DNLS operator, invariant under Z 2 having poles at degenerate points and we construct solutions to the entwining Yang-Baxter equation.The paper is organised as follows. In the next section, we provide preliminary material, definitions and we fix the notation. In particular, we give the definition of the entwining Yang-Baxter equation and we explain the relation between entwining Yang-Baxter maps and matrix refactorisation problems. For entwining maps defined by matrix refactorisation problems, we discuss their birationality, we prove a statement regarding and their invariants and we give the definition of their integrability in the Liouville sense. Finally, we list all the Darboux matrices we are using throughout the text. In...