  We study the effects of non-standard self-interactions (NSSI) of neutrinos
streaming out of a core-collapse supernova. We show that with NSSI, the
standard linear stability analysis gives rise to linearly as well as
exponentially growing solutions. For a two-box spectrum, we demonstrate
analytically that flavor-preserving NSSI lead to a suppression of bipolar
collective oscillations. In the intersecting four-beam model, we show that
flavor-violating NSSI can lead to fast oscillations even when the angle between
the neutrino and antineutrino beams is obtuse, which is forbidden in the
Standard Model. This leads to the new possibility of fast oscillations in a
two-beam system with opposing neutrino-antineutrino fluxes, even in the absence
of any spatial inhomogeneities. Finally, we solve the full non-linear equations
of motion in the four-beam model numerically, and explore the interplay of fast
and slow flavor conversions in the long-time behavior, in the presence of NSSI.
