  After the first predictions in seventies that in dense nuclear matter, for
example, in neutron stars there could be pion condensation phenomenon, it was
shown that at least s-wave pion condensation is unlikely to occur in such a
medium in different approaches including NJL model consideration with
neutrality and $\beta$-equilibrium condition. Then it has been found a
condition that can promote this phenomenon in dense baryonic matter, it is
chiral imbalance, so now it is interesting if this generation can survive
rather strict requirement of electric neutrality and $\beta$-equilibrium. Here
in this paper the generation of pion condensation in dense baryonic matter by
chiral imbalance is investigated. It is shown that electric neutrality and
$\beta$-equilibrium conditions do not spoil this phenomenon and this leads to
interesting applications for physics of neutron stars. There are several
possible mechanisms of generation of chiral imbalance in dense matter. In a
view of latest and forthcoming NICER results and the first observed and
possibly new events of neutron star mergers it is rather interesting to explore
how possible chiral imbalance in neutron star can influence and change EOS and,
hence, mass-radius relation. It is a first step in that direction.
