  The flavor evolution of a neutrino gas can show ''slow'' or ''fast''
collective motion. In terms of the usual Bloch vectors to describe the
mean-field density matrices of a homogeneous neutrino gas, the slow two-flavor
equations of motion (EOMs) are
$\dot{\mathbf{P}}_\omega=(\omega\mathbf{B}+\mu\mathbf{P})\times\mathbf{P}_\omega$,
where $\omega=\Delta m^2/2E$, $\mu=\sqrt{2} G_{\mathrm{F}}
(n_\nu+n_{\bar\nu})$, $\mathbf{B}$ is a unit vector in the mass direction in
flavor space, and $\mathbf{P}=\int d\omega\,\mathbf{P}_\omega$. For an
axisymmetric angle distribution, the fast EOMs are
$\dot{\mathbf{D}}_v=\mu(\mathbf{D}_0-v\mathbf{D}_1)\times\mathbf{D}_v$, where
$\mathbf{D}_v$ is the Bloch vector for lepton number, $v=\cos\theta$ is the
velocity along the symmetry axis, $\mathbf{D}_0=\int dv\,\mathbf{D}_v$, and
$\mathbf{D}_1=\int dv\,v\mathbf{D}_v$. We discuss similarities and differences
between these generic cases. Both systems can have pendulum-like instabilities
(soliton solutions), both have similar Gaudin invariants, and both are
integrable in the classical and quantum case. Describing fast oscillations in a
frame comoving with $\mathbf{D}_1$ (which itself may execute pendulum-like
motions) leads to transformed EOMs that are equivalent to an abstract slow
system. These conclusions carry over to three flavors.
