  The effective many-body Hamiltonian which describes vacuum oscillations and
self interactions of neutrinos in a two flavor mixing scheme under the single
angle approximation has the same dynamical symmetries as the well known BCS
pairing Hamiltonian. These dynamical symmetries manifest themselves in terms of
a set of constants of motion and can be useful in formulating the collective
oscillation modes in an intuitive way. In particular, we show that a neutrino
spectral split can be simply viewed as an avoided level crossing between the
eigenstates of a mean field Hamiltonian which includes a Lagrange multiplier in
order to fix the value of an exact many-body constant of motion. We show that
the same dynamical symmetries also exist in the three neutrino mixing scheme by
explicitly writing down the corresponding constants of motion.
