  The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization in
terms of a two-level boson pairing Hamiltonian. Through this realization, it
has 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 of
the model tostudy the behavior of the spectral parameters (pairons) that
completely determine the wave function in the different phases, and across the
phase transitions. Based on the relation between the Richardson equations and
the Lam\'e differential equations we develop a method to obtain numerically the
pairons. The dynamics of pairons in the ground and excited states provides new
insights into the first, second and third order phase transitions, as well as
into the crossings taking place in the LMG spectrum.
