From the Kovalevskaya to the Lagrange case in rigid body motion

We study the motion of a family of symmetric tops in which the center of mass is located between the symmety plane and the symmetry axis of the inertia matrix. We analyze the transition from the Kovalevskaya to the Lagrange integrable cases using Poincaré sections and symmetry lines. The fate of periodic orbits as a function of the location of the top's center of mass is analyzed. The critical points of the Kovalevskaya constant are calculated in terms of the energy, of the angular momentum about the vertical, and of the Kovalevskaya constant itself.