= Symmetric top under constant transverse body torque
{title2=$\dot z=i\nu z+(\tau_1+i\tau_2)/I_\perp$}
Let a <symmetric top> have <principal moments of inertia> $I_1=I_2=I_\perp$ and $I_3=I_\parallel$. Apply a <torque> $(\tau_1,\tau_2,0)$ constant in a <principal body frame>. Then the axial <angular velocity> $c=\omega_3$ is a <conserved quantity>. Define $z=\omega_1+i\omega_2$ and $\nu=(I_\parallel-I_\perp)c/I_\perp$. The first two of <Euler's equations (rigid body dynamics)> combine into the equation in the title. This derivation uses constancy in the <principal body frame>, which generally does not mean constant <torque> in an <inertial frame>.
Multiplying by the <integrating factor> $e^{-i\nu t}$ proves, for $\nu\ne0$,
$$
z(t)=z(0)e^{i\nu t}+\frac{\tau_1+i\tau_2}{i\nu I_\perp}(e^{i\nu t}-1).
$$
Equivalently $z$ rotates about the constant point $z_* =i(\tau_1+i\tau_2)/(\nu I_\perp)$ in the complex transverse plane. When $\nu=0$, the solution is instead $z(t)=z(0)+(\tau_1+i\tau_2)t/I_\perp$. The <angular velocity> is $(\operatorname{Re}z,\operatorname{Im}z,c)$.
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