  We propose a scheme of searches for the Abelian $Z'$ gauge boson in the
Drell-Yan process at $\sqrt{S}=1.96$ TeV. We base our considerations on
renor\-ma\-li\-za\-tion-group relations between the $Z'$ couplings to
standard-model fermions. Considering the range of energies near the $Z$-boson
peak, namely $66-116$ GeV for the invariant mass of a leptonic pair, we propose
an integration scheme to construct two-parametric observables suitable for $Z'$
searches in the $p\bar{p} \to l^+ l^-$ scattering. The observables allow to
constrain the $Z'$ vector and axial-vector couplings to SM fermions in a
general phenomenological parameterization with non-universal $Z'$ interactions
with fermion generations. We also consider the cases of generation-universal
$Z'$ boson and leptophobic $Z'$ boson, and show that one-parametric observables
exist for these scenarios. The research is aimed to supplement direct searches
for $Z'$ as an on-shell state in a specific set of new-physics models.
