= Solution
<Noether theorem> associates a conserved quantity $\sum_i p_i\delta q_i$ to a continuous symmetry of the <Lagrangian> (with the usual boundary-term correction if the Lagrangian changes by a total derivative). The angles $\psi$ and $\phi$ are <cyclic coordinates>, so for the <heavy symmetric top>
$$
\boxed{p_\psi=I_3(\dot\psi+\dot\phi\cos\theta)=b,\qquad p_\phi=I_1\dot\phi\sin^2\theta+b\cos\theta=a.}
$$
Time-translation symmetry gives conserved <energy>, whose <Hamiltonian> expression is
$$
\boxed{E=\frac{p_\theta^2}{2I_1}+\frac{(p_\phi-p_\psi\cos\theta)^2}{2I_1\sin^2\theta}+\frac{p_\psi^2}{2I_3}+Mgl\cos\theta.}
$$
It is quadratic in the momenta; $p_\theta=I_1\dot\theta$. Eliminating the cyclic velocities and putting $\mu=\cos\theta$, so $\dot\mu^2=\sin^2\theta\dot\theta^2$, yields
$$
\boxed{\dot\mu^2=\frac2{I_1}(1-\mu^2)\left(E-\frac{b^2}{2I_3}-Mgl\mu\right)-\frac{(a-b\mu)^2}{I_1^2}=F(\mu).}
$$
The first term has degree three and the second degree two, displaying the cubic governing <nutation>.
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