An integrable family of torqued, damped, rigid rotors

Expositions of the Euler equations for rotation a rigid body often invoke idea specially damped system whose energy dissipates while its angular momentum magnitude is conserved in frame. An attempt to explicitly construct such damping function leads more general, but still integrable, cubic trajectories are confined nested sets quadric surfaces space. For some choices parameters, lines fixed points along both largest and smallest moment inertia axes can be simultaneously attracting. Limiting cases those that conserve either or momentum. Parallels with rod mechanics, micromagnetics, particles effective mass briefly discussed.