Automorphic symmetries and $ AdS_n $ integrable deformations

We develop the method based on $\mathcal{B}$-automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces. We initiate the technique by implementing it to the two-dimensional models and resolve classification problem, which not only confirms known vertex model solution space, but also extends to the new $\mathfrak{sl}_{2}$ deformed sector. The generalisation of the approach for string integrable models is provided and allows to find new integrable deformations and associated $R$-matrices. Hence novel integrable solutions appear to be of non-difference form that admit $AdS_2$ and $AdS_3$ $S$-matrices as special cases, we also obtain embedding of double deformed sigma model $R$-matrix into one of the solutions. The braiding and crossing for the found models as well as their emergence with respect to the deformation parameter $k$ are shown.