“Twisted” rational $r$-matrices and the algebraic Bethe ansatz: Applications to generalized Gaudin models, Bose–Hubbard dimers, and Jaynes–Cummings–Dicke-type models

We construct quantum integrable systems associated with the Lie algebra $gl(n)$ and non-skew-symmetric "shifted and twisted" rational $r$-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, $n$-level $(n-1)$-mode Jaynes–Cummings–Dicke-type models in the $\Lambda$-configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz.