Gaudin-type models, non-skew-symmetric classical r-matrices and nested Bethe ansatz

We consider quantum integrable systems associated with the Lie algebra gl(n)gl(n) and Cartan-invariant non-dynamical non-skew-symmetric classical r-matrices. We describe the sub-class of Cartan-invariant non-skew-symmetric r  -matrices for which exists the standard procedure of the nested Bethe ansatz associated with the chain of embeddings gl(n)⊃gl(n−1)⊃gl(n−2)⊃⋯⊃gl(1)gl(n)⊃gl(n−1)⊃gl(n−2)⊃⋯⊃gl(1). We diagonalize the corresponding quantum integrable systems by its means. We illustrate the obtained results by the examples of the generalized Gaudin systems with and without external magnetic field associated with three classes of non-dynamical non-skew-symmetric classical r-matrices.