  In dense neutrino backgrounds present in supernovae and in the early Universe
neutrino oscillations may exhibit complex collective phenomena, such as
synchronized oscillations, bipolar oscillations and spectral splits and swaps.
We consider in detail possible decoherence effects on the simplest of these
phenomena -- synchronized neutrino oscillations that can occur in a uniform and
isotropic neutrino gas. We develop an exact formalism of spectral moments of
the flavour spin vectors describing such a system and then apply it to find
analytical approaches that allow one to study decoherence effects on its
late-time evolution. This turns out to be possible in part due to the existence
of the (previously unknown) exact conservation law satisfied by the quantities
describing the considered neutrino system. Interpretation of the decoherence
effects in terms of neutrino wave packet separation is also given, both in the
adiabatic and non-adiabatic regimes of neutrino flavour evolution.
