  In core-collapse supernovae and neutron star mergers, the neutrino density is
so large that neutrino-neutrino refraction can lead to flavor conversion, if a
zero-crossing is present in the neutrino flavor lepton number (FLN) angular
distribution and the neutrino self-interaction strength $\mu=\sqrt{2} G_F
n_\nu$ represents the characteristic timescale of the system. It has been
empirically realized that the vacuum frequency $\omega=\Delta m^2/2E$ affects
the development of flavor conversion in the presence of zero-crossing even if
$\omega \ll \mu$. Focusing on a homogeneous and axially symmetric neutrino gas,
we explore the role of $\omega$ in the onset of flavor instabilities. We find
that a non-zero vacuum frequency can be responsible for inducing flavor
instabilities even when the neutrino self-interaction strength is much larger
than the vacuum frequency. Moreover, mapping a neutrino ensemble with $\omega
\neq 0$ into an effective system with $\omega =0$, we find that a system with
no FLN zero-crossing can effectively develop one for $\omega \neq 0$ becoming
unstable.
