  We consider the flavor evolution of a dense neutrino gas by taking into
account both vacuum oscillations and self interactions of neutrinos. We examine
the system from a many-body perspective as well as from the point of view of an
effective one-body description formulated in terms of the neutrino polarization
vectors. We show that, in the single angle approximation, both the many-body
picture and the effective one-particle picture possess several constants of
motion. We write down these constants of motion explicitly in terms of the
neutrino isospin operators for the many-body case and in terms of the
polarization vectors for the effective one-body case. The existence of these
constants of motion is a direct consequence of the fact that the collective
neutrino oscillation Hamiltonian belongs to the class of Gaudin Hamiltonians.
This class of Hamiltonians also includes the (reduced) BCS pairing Hamiltonian
describing superconductivity. We point out the similarity between the
collective neutrino oscillation Hamiltonian and the BCS pairing Hamiltonian.
The constants of motion manifest the exact solvability of the system. Borrowing
the well established techniques of calculating the exact BCS spectrum, we
present exact eigenstates and eigenvalues of both the many-body and the
effective one-particle Hamiltonians describing the collective neutrino
oscillations. For the effective one-body case, we show that spectral splits of
neutrinos can be understood in terms of the adiabatic evolution of some
quasi-particle degrees of freedom from a high density region where they
coincide with flavor eigenstates to the vacuum where they coincide with mass
eigenstates. We write down the most general consistency equations which should
be satisfied by the effective one-body eigenstates and show that they reduce to
the spectral split consistency equations for the appropriate initial
conditions.
