  Self-induced flavor conversions of supernova (SN) neutrinos have been
characterized in the spherically symmetric "bulb" model, reducing the neutrino
evolution to a one dimensional problem along a radial direction. We lift this
assumption, presenting a two-dimensional toy-model where neutrino beams are
launched in many different directions from a ring. We find that
self-interacting neutrinos spontaneously break the spatial symmetries of this
model. As a result the flavor content and the lepton number of the neutrino gas
would acquire seizable direction-dependent variations, breaking the coherent
behavior found in the symmetric case. This finding would suggest that the
previous results of the self-induced flavor evolution obtained in
one-dimensional models should be critically re-examined.
