  In the L_{\mu} - L_{\tau} model the 3.6$\sigma$ discrepancy between the
predicted and measured values of the anomalous magnetic moment of positive
muons can be explained by the existence of a new dark boson Z' with a mass in
the sub-GeV range, which is coupled at tree level predominantly to the second
and third lepton generations. However, at the one-loop level, the Z' coupling
to electrons or quarks can be induced via the \gamma -Z' kinetic mixing, which
is generated through the loop involving the muon and tau lepton. This loophole
has important experimental consequences since it opens up new possibilities, in
particular for the complementary searches of the Z' in the ongoing NA64 and
incoming dark photon experiments with high-energy electrons. An extension of
the model able to explain relic Dark Matter density is also discussed.
