= Solution
The two cyclic <Euler angles> give conserved <canonical momenta>
$$
L_3=p_\psi=I_3(\dot\psi+\dot\phi\cos\theta),\qquad L_z=p_\phi=I_1\dot\phi\sin^2\theta+L_3\cos\theta.
$$
Here $L_3$ is the body-axis <angular momentum> and $L_z$ the vertical <angular momentum>. Thus \b[$\omega_3=L_3/I_3$ is constant]. Since the <Lagrangian> has no explicit time dependence, the <energy> is also conserved:
$$
E=\frac{I_1}2\dot\theta^2+\frac{I_1}2\dot\phi^2\sin^2\theta+\frac{L_3^2}{2I_3}+Mgl\cos\theta.
$$
These are the three independent conserved quantities furnished by the continuous symmetries of the <heavy symmetric top>.
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