  Neutrino self-interactions are known to lead to non-linear collective flavor
oscillations in a core-collapse supernova. We point out new possible effects of
non-standard self-interactions (NSSI) of neutrinos on flavor conversions in a
two-flavor framework. We show that, for a single-energy neutrino-antineutrino
ensemble, a flavor instability is generated even in normal hierarchy for large
enough NSSI. Using a toy model for the neutrino spectra, we show that
flavor-preserving NSSI lead to pinching of spectral swaps, while
flavor-violating NSSI cause swaps to develop away from a spectral crossing or
even in the absence of a spectral crossing. Consequently, NSSI could give rise
to collective oscillations and spectral splits even during neutronization
burst, for both hierarchies.
