  The enhancement mechanism due to the resonant properties of the $\rho$ and
$\omega$ mesons, which are close in mass, are analysed when such resonances
carry different momenta. Considerations from the particular process where they
appear to the individual resonant features are at play for the appearance of
the global resonant manifestation. In this work, we first consider the $e^+e^-
\to \pi^0 \pi^0 \gamma$ process. We use the differential cross section at a
given angle of emission of one of the pions, to tune the individual features of
the two resonances and exhibit how both resonances combine to produce the
enhancement. Then, we incorporate the $\rho^\prime$ using the information
obtained from the $ e^+e^- \to \pi^0 \pi^0 \gamma$ total scattering process and
show that, it becomes important thanks to the same enhancement mechanism
between the $\rho$ and the $\omega$. In a second step, we use a similar
approach to describe a model dependent contribution to the $\tau^- \to
\pi^-\pi^0\nu_\tau\gamma$ decay, the so-called $\omega$ channel. We show that
the dipion invariant mass distribution at particular angles is sensitive to the
individual resonant states. We compute the interference of this channel with
the known dominant model independent contribution, and show how a better
knowledge of the $e^+e^- \to \pi^0 \pi^0 \gamma$ process can help to properly
account for such model dependent effects. The implication on the isospin
symmetry breaking correction to tau-based estimates of the muon magnetic dipole
moment is assessed.
