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The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization interms of a two-level boson pairing Hamiltonian. Through this realization, ithas been shown that the LMG model is a particular case of the SU (1, 1)Richardson-Gaudin (RG) integrable models. We exploit the exact solvability ofthe model tostudy the behavior of the spectral parameters (pairons) thatcompletely determine the wave function in the different phases, and across thephase transitions. Based on the relation between the Richardson equations andthe Lam\'e differential equations we develop a method to obtain numerically thepairons. The dynamics of pairons in the ground and excited states provides newinsights into the first, second and third order phase transitions, as well asinto the crossings taking place in the LMG spectrum.

The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1) Richardson-Gaudin integrable models