  The Martin-Schwinger hierarchy of correlations are reexamined and the
three-particle correlations are investigated under various partial summations.
Besides the known approximations of screened, ladder and maximally crossed
diagrams the pair-pair correlations are considered. It is shown that the
recently proposed asymmetric Bethe-Salpeter equation to avoid unphysical
repeated collisions is derived as a result of the hierarchical dependencies of
correlations. Exceeding the parquet approximation we show that an asymmetry
appears in the selfconsistent propagators. This form is superior over the
symmetric selfconsistent one since it provides the Nambu-Gorkov equations and
gap equation for fermions and the Beliaev equations for bosons while from the
symmetric form no gap equation results. The selfenergy diagrams which account
for the subtraction of unphysical repeated collisions are derived from the
pair-pair correlation in the three-particle Greenfunction. It is suggested to
distinguish between two types of selfconsistency, the channel-dressed
propagators and the completely dressed propagators, with the help of which the
asymmetric expansion completes the Ward identity and is $\Phi$-derivable.
