Generalized Riemann-Hilbert-Birkhoff decomposition and a new class of higher grading integrable hierarchies

We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations. Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models). For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism.