  We address the novel structures arising in quantum and string integrable
theories, as well as construct methods to obtain them and provide further
analysis. Specifically, we implement the automorphic symmetries on periodic
lattice systems to obtain integrable hierarchies, whose commutativity and
integrable transformations induce a generating structure of integrable classes.
This prescription is first applied to 2-dim and 4-dim setups, where we find the
new $ \mathfrak{sl}_{2} $ sector, $ \mathfrak{su}(2) \oplus \mathfrak{su}(2) $
with superconductive modes, Generalised Hubbard type classes and more. The
corresponding 2- and 4-dim $ R $ matrices are resolved through perturbation
theory, that allows to recover an exact result. We then construct a boost
recursion that allows to address the systems, whose $ R $-/$ S $-matrices
exhibit arbitrary spectral dependence, that also is an apparent property of the
scattering operators in $ AdS $ integrability. It is then possible to implement
the last for Hamiltonian Ans\"atze in $ D = 2,3,4 $, which leads to new models
in all dimensions. We also provide a method based on a coupled differential
system that allows to resolve for $ R $ matrices exactly. Important it is
possible isolate a special class of models of non-difference form in 2-dim case
(6vB/8vB), which provides a new structure consistently arising in $ AdS_{3} $
and $ AdS_{2} $ string backgrounds. We prove that these classes can be
represented as deformations of the $ AdS_{\{ 2,3 \}} $ models. We also work out
that the latter satisfy free fermion constraint, braiding unitarity, crossing
and exhibit deformed algebraic structure that shares certain properties with $
AdS_{3} \times S^{3} \times \mathcal{M}^{4} $ and $ AdS_{2} \times S^{2} \times
T^{6} $ models. The embedding and mappings of 6vB/8vB deformations are
demonstrated, a discussion on relation to sigma model candidates is provided.
