Motion of a disk within a sphere

The motion of a round dynamically symmetrical disk within a sphere covered by smooth ice is considered. It is shown that in the absence of a gravity field equations of motion may be reduced to quadratures. Conditions for the stability of stationary motions have been found. Probability of a falling of a disk has been investigated. Regions allowed for the motion in a gravity field and characterizing by the disk orientation and the situation of a trajectory of a contact point on a sphere surface have been built. The conclusion on the non-integrability on the analysis of Poincare sections of a problem on a round dynamically symmetrical disk moving on a smooth sphere surface has been made.