Generalized Gaudin systems in a magnetic field and non-skew-symmetric r-matrices

Abstract

We construct integrable cases of generalized classical and quantum Gaudin spin chains in an external magnetic field. For this purpose, we generalize the ‘shift of argument method’ onto the case of classical and quantum integrable systems governed by an arbitrary -valued, non-dynamical classical r-matrix with spectral parameters. We consider several examples of the obtained construction for the cases of skew-symmetric, ‘twisted’ non-skew-symmetric and ‘anisotropic’ non-skew-symmetric classical r-matrices. We show, in particular, that in a general case in order for the Gaudin system in a magnetic field to be integrable, the corresponding magnetic field should be non-homogeneous.