Novel integrability in string theory from automorphic symmetries

We develop a technique based on the boost automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces. We initiate the method by implementing it in two-dimensional models and resolve a classification problem, which not only confirms the known vertex model solution space but also extends to the new $\mathfrak{sl}_2$ deformed sector. A generalization of the approach to integrable string backgrounds is provided and allows finding new integrable deformations and associated $R$-matrices. The new integrable solutions appear to be of a nondifference or pseudo-difference form admitting $AdS_2$ and $AdS_3$ $S$-matrices as special cases (embeddings), which also includes a map of the double-deformed sigma model $R$-matrix. The corresponding braiding and conjugation operators of the novel models are derived. We also demonstrate implications of the obtained free-fermion analogue for $AdS$ deformations.