Dynamics of dust near the gravitating center. The integrals of motion

A classical model of dust dynamics near a gravitating center in a rotating frame of reference is considered. The problem is solved by the method of characteristics. The integrals of motion on the characteristics are calculated in explicit form and their role is determined for various types of formulation of the problem. Specific examples are given of constructing solutions for some simple problems, in particular, for ring structures in dust disks and a formal model for the formation of jets.