Exact solutions for eigenvalues and eigenfunctions of some nuclear many-body systems are found by using an infinite-dimensional, Lie-algebraic approach based on the corresponding Bethe ansatz. Applications of the theory, including solutions of some nuclear pairing problems and U(5)<->SO(6) transitional Hamiltonians of the interacting boson model, are given. The relationship between this new method and other Bethe ansatz solutions in completely integrable systems is discussed.
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