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