= Solution
When $a=b<c$, the ellipsoid is an axisymmetric rigid body with
$$
I_1=I_2=\frac M5(a^2+c^2)>I_3=\frac{2M}{5}a^2.
$$
The angle between its symmetry axis and the fixed angular momentum is constant. In an inertial frame the symmetry axis precesses uniformly around $\mathbf L$, while the body simultaneously spins about that axis. The angular-velocity vector also traces a circular cone about $\mathbf L$. The limiting cases are pure rotation about the symmetry axis and pure rotation about a perpendicular principal axis.
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