The Goryachev integral for the equations of motion of a complex mechanical system

The motion of a mechanical system with a time-varying mass value and a mass distribution configuration is considered, in which the invariant basis of the system (the carrier body) moves around a fixed pole. The motion occurs in a homogeneous field of gravity under the action of reactive forces due to the mass change of the system. A criterion for the existence of an independent first integral of the system of equations of motion, which is a generalized analogue of the classical D.N. Goryachev integral.