Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl(2)⊗sl(2)-valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes-Cummings-Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes-Cummings-Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.