  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 Schr\"odinger
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.
