Lagrange, Euler, and Kovalevskaya tops

In classical mechanics, the rotation of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three famous cases that are integrable, the Euler, the Lagrange, and the Kovalevskaya top, which are in fact the only integrable cases when the system is subject to holonomic constraints. In addition to the energy, each of these tops involves two additional constants of motion that give rise to the integrability.

The Euler top describes a free top without any particular symmetry moving in the absence of any external torque, and for which the fixed point is the center of gravity. The Lagrange top is a symmetric top, in which two moments of inertia are the same and the center of gravity lies on the symmetry axis. The Kovalevskaya top is a special symmetric top with a unique ratio of the moments of inertia which satisfy the relation

${\displaystyle I_{1}=I_{2}=2I_{3},}$

That is, two moments of inertia are equal, the third is half as large, and the center of gravity is located in the plane perpendicular to the symmetry axis (parallel to the plane of the two equal points).