  We formulate the general form of omega-Z-gamma vertex in the framework based
on the hidden local symmetry (HLS), which arises from the gauge invariant terms
for intrinsic parity-odd (IP-odd) part of the effective action. Those terms are
given as the homogeneous part of the general solution (having free parameters)
to the Wess-Zumino (WZ) anomaly equation and hence are not determined by the
anomaly, in sharp contrast to the Harvey-Hill-Hill (HHH) action where the
relevant vertex is claimed to be uniquely determined by the anomaly. We show
that, even in the framework that HHH was based on, the omega-Z-gamma vertex is
actually not determined by the anomaly but by the homogeneous (anomaly-free)
part of the general solution to the WZ anomaly equation having free parameters
in the same way as in the HLS formulation: The HHH action is just a particular
choice of the free parameters in the general solution. We further show that the
omega-Z-gamma vertex related to the neutrino (nu) - nucleon (N) scattering
cross section sigma(nu N -> nu N (N')gamma) is determined not by the anomaly
but by the anomaly-free part of the general solution having free parameters.
Nevertheless we find that the cross section sigma(nu N -> nu N(N')gamma) is
related through the Ward-Takahashi identity to Gamma(omega -> pi^0 gamma) which
has the same parameter-dependence as that of sigma(nu N -> nu N(N')gamma) and
hence the ratio sigma(nu N -> nu N(N')gamma)/Gamma(omega -> pi^0 gamma) is
fixed independently of these free parameters. Other set of the free parameters
of the general solution can be fixed to make the best fit of the omega -> pi^0
l^+ l^- process, which substantially differs from the HHH action. This gives a
prediction of the cross section sigma(nu N -> nu N (N')gamma^*(l^+ l^-)) to be
tested at nu-N collision experiments in the future.
