  It has been recently discovered that removing the axial symmetry in the
"multi-angle effects" associated with the neutrino-neutrino interactions for
supernova (SN) neutrinos, a new multi-azimuthal-angle (MAA) instability would
trigger flavor conversions in addition to the ones caused by the bimodal and
multi-zenith-angle (MZA) instabilities. We investigate the dependence of the
MAA instability on the original SN neutrino fluxes, performing a stability
analysis of the linearized neutrino equations of motion. We compare these
results with the numerical evolution of the SN neutrino non-linear equations,
looking at a local solution along a specific line of sight, under the
assumption that the transverse variations of the global solution are small. We
also assume that self-induced conversions are not suppressed by large matter
effects. We show that the pattern of the spectral crossings (energies where
F_{\nu_e} = F_{\nu_x}, and F_{\bar\nu_e} = F_{\bar\nu_x}) is crucial in
determining the impact of MAA effects on the flavor evolution. For neutrino
spectra with a strong excess of \nu_e over \bar\nu_e, presenting only a
single-crossing, MAA instabilities would trigger new flavor conversions in
normal mass hierarchy. In our simplified flavor evolution scheme, these would
lead to spectral swaps and splits analogous to what produced in inverted
hierarchy by the bimodal instability. Conversely, in the presence of spectra
with a moderate flavor hierarchy, having multiple crossing energies, MZA
effects would produce a sizable delay in the onset of the flavor conversions,
inhibiting the growth of the MAA instability. In this case the splitting
features for the oscillated spectra in both the mass hierarchies are the ones
induced by the only bimodal and MZA effects.
