  We introduce a general scheme to consistently truncate equations of motion
for Green's functions. Our scheme is guaranteed to generate physical Green's
functions with real excitation energies and positive spectral weights. There
are free parameters in our scheme akin to mean field parameters that may be
determined to get as good an approximation to the physics as possible. As a
test case we apply our scheme to a two-pole approximation for the 2D Hubbard
model. At half-filling we find an insulating solution with several interesting
properties: it has low expectation value of the energy and it gives upper and
lower Hubbard bands with the full non-interacting bandwidth in the large U
limit. Away from half-filling, in particular in the intermediate interaction
regime, our scheme allows for several different phases with different number of
Fermi surfaces and topologies.
