Exactly-solvable models derived from a generalized Gaudin algebra

We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen-Cooper-Schrieffer, Suhl-Matthias-Walker, Lipkin-Meshkov-Glick, the generalized Dicke and atom-molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.