We formulate the general form of ω-Z-γ 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 ω-Z-γ 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 ω-Z-γ vertex related to the neutrino (ν) -nucleon (N ) scattering cross section σ(νN → νN (N ′ )γ) 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 σ(νN → νN (N ′ )γ) is related through the Ward-Takahashi identity to Γ(ω → π 0 γ) which has the same parameter-dependence as that of σ(νN → νN (N ′ )γ) and hence the ratio σ(νN → νN (N ′ )γ)/Γ(ω → π 0 γ) 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 ω → π 0 l + l − process, which substantially differs from the HHH action. This gives a prediction of the cross section σ(νN → νN (N ′ )γ * (l + l − )) to be tested at ν-N collision experiments in the future.