  It was recently shown that left-right symmetric models for elementary
particles can be built with only two Higgs doublets. The general consequence of
these models is that the left and right fermionic sectors can be connected by a
new neutral gauge boson $Z'$ having its mass as the only additional new
parameter. In this paper we study the influence of the fundamental fermionic
representation for this new neutral gauge boson. Signals of possible new heavy
neutral gauge bosons are investigated for the future electron-positron
colliders at $\sqrt s = 500$ GeV, 1 TeV and 3 TeV. The total cross sections,
forward-backward and left-right asymmetries and model differences are
calculated for the process $e^+ e^- \longrightarrow \mu^+ \mu^-$. Bounds on
$Z'$ masses are estimated.
