Non-skew-symmetric classical r-matrices and integrable “image” proton–neutron BCS models Original Research Article

We study integrable cases of the pairing BCS Hamiltonians containing several types of fermions and possessing non-uniform coupling constants. We prove that there exist three classes of such the integrable models associated with “image-graded” non-skew-symmetric classical r-matrices with spectral parameters and Lie algebras image, image and image, respectively. The proposed models are higher rank generalizations of the so-called “image” one-type fermion (image) BCS model. In the partial case of two types of fermions (image) the obtained models may be interpreted as image, “image” proton–neutron integrable models. In particular, in the case of image we obtain the “image”-analogue of the famous integrable proton–neutron model of Richardson. We find the spectrum of the constructed Hamiltonians in terms of solutions of Bethe-type equations.