  The two-dimensional Hubbard model is analyzed in the framework of the
two-pole expansion. It is demonstrated that several theoretical approaches,
when considered at their lowest level, are all equivalent and share the
property of satisfying the conservation of the first four spectral momenta. It
emerges that the various methods differ only in the way of fixing the internal
parameters and that it exists a unique way to preserve simultaneously the Pauli
principle and the particle-hole symmetry. A comprehensive comparison with
respect to some general symmetry properties and the data from quantum Monte
Carlo analysis shows the relevance of imposing the Pauli principle.
