  In this paper we study in detail the deformations introduced in [1] of the
integrable structures of the AdS$_{2,3}$ integrable models. We do this by
embedding the corresponding scattering matrices into the most general solutions
of the Yang-Baxter equation. We show that there are several non-trivial
embeddings and corresponding deformations. We work out crossing symmetry for
these models and study their symmetry algebras and representations. In
particular, we identify a new elliptic deformation of the $\rm AdS_3 \times S^3
\times M^4$ string sigma model.
