Solution
= Solution
Use the conserved $p_\phi=p_\psi=I_3\omega$ in the canonical-momentum equation. For a nonzero tilt,
$$
I_1\dot\phi\sin^2\theta=I_3\omega(1-\cos\theta).
$$
Cancelling with the half-angle identities gives
$$
\boxed{\dot\phi=\frac{I_3\omega}{I_1(1+\cos\theta)}
=\frac{C}{2\cos^2(\theta/2)}.}
$$
Its regular limit at the north pole is $C/2$, although the azimuth itself is not a uniquely defined coordinate at that pole.