  The Lipkin-Meshkov-Glick (LMG) model was devised to test the validity of
different approximate formalisms to treat many-particle systems. The model was
constructed to be exactly solvable and yet non-trivial, in order to capture
some of the main features of real physical systems. In the present
contribution, we explicitly review the fact that different many-body
approximations commonly used in different fields in physics clearly fail to
describe the exact LMG solution. With similar assumptions as those adopted for
the LMG model, we propose a new Hamiltonian based on a general two-body
interaction. The new model (Extended LMG) is not only more general than the
original LMG model and, therefore, with a potentially larger spectrum of
applicability, but also the physics behind its exact solution can be much
better captured by common many-body approximations. At the basis of this
improvement lies a new term in the Hamiltonian that depends on the number of
constituents and polarizes the system; the associated symmetry breaking is
discussed, together with some implications for the study of more realistic
systems.
