  We consider collective oscillations of neutrinos, which are emergent
nonlinear flavor evolution phenomena instigated by neutrino-neutrino
interactions in astrophysical environments with sufficiently high neutrino
densities. We investigate the symmetries of the problem in the full three
flavor mixing scheme and in the exact many-body formulation by including the
effects of CP violation and neutrino magnetic moment. We show that, similar to
the two flavor scheme, several dynamical symmetries exist for three flavors in
the single-angle approximation if the net electron background in the
environment and the effects of the neutrino magnetic moment are negligible.
Moreover, we show that these dynamical symmetries are present even when the CP
symmetry is violated in neutrino oscillations. We explicitly write down the
constants of motion through which these dynamical symmetries manifest
themselves in terms of the generators of the SU(3) flavor transformations. We
also show that the effects due to the CP-violating Dirac phase factor out of
the many-body evolution operator and evolve independently of nonlinear flavor
transformations if neutrino electromagnetic interactions are ignored. In the
presence of a strong magnetic field, CP-violating effects can still be
considered independently provided that an effective definition for neutrino
magnetic moment is used.
