  A modified $GW$ approximation to many - body systems is developed. The
approximation has the same computational complexity as the traditional $GW$
approach, but uses a different truncation scheme. This scheme neglects high
order connected correlation functions. A covariant (preserving Ward identities
due to charge conservation) scheme for two - body correlators is employed,
which holds the relation between the charge correlator and charge
susceptibility. The method is tested on the two - dimensional one - band
Hubbard model. Results are compared with exact diagonalization, the fluctuation
- exchange (FLEX) theory and determinantal quantum Monte Carlo (DQMC) approach.
The comparison for the (one - body) Green's function demonstrates that it is
more precise in strong - coupling regime (especially away from half - filling)
than similar - complexity approximations $GW$ or FLEX. The charge correlator is
in excellent agreement with the numerically exact result obtained from DQMC.
