Spin chains in magnetic field,non-skew-symmetric classical r-matrices and BCS-type integrable systems

We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)$\mathit{so}(3)$-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r  -matrices diagonal in the sl(2)$\mathit{sl}(2)$ basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.