In this paper, the relationship between the classical Dicke-Jaynes-Cummings-Gaudin (DJCG) model and the nonlinear Schrödinger (NLS) equation is studied. It is shown that the classical DJCG model is equivalent to a stationary NLS equation. Moreover, the standard NLS equation can be solved by the classical DJCG model and a suitably chosen higher order flow. Further, it is also shown that classical DJCG model can be transformed into the classical Gaudin spin model in an external magnetic field through a deformation of Lax matrix. Finally, the separated variables are constructed on the common level sets of Casimir functions and the generalized action-angle coordinates are introduced via the Hamilton-Jacobi equation.

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On the basis of two sets of Lenard recursion sequences and zero-curvature equation associated with a matrix spectral problem, we derive the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.

<abstract><p>The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.</p></abstract>