Quartic equation for one component of torque-free angular velocity
= Quartic equation for one component of torque-free angular velocity
Eliminating $\omega_1^2$ and $\omega_2^2$ between the conserved energy and squared angular momentum gives
$$
\dot\omega_3^2=-\frac{
\left[L^2-2EI_2+I_3(I_2-I_3)\omega_3^2\right]
\left[L^2-2EI_1+I_3(I_1-I_3)\omega_3^2\right]
}{I_1I_2I_3^2}.
$$
The right-hand side is a quartic polynomial in $\omega_3$; its real turning points delimit the nutation of the body-frame angular velocity.