A hinged joint is formed on a volumetric rotary machine with a spherical working cavity formed between a spherical rotor part and the spherical surface of the body, between a piston, which is mounted in such a way as to enable rotary oscillations relative to the rotor in the plane passing along the axis of rotation of the rotor, and a rotor, which is mounted in such a way as to enable rotation about the spherical rotor part. This solution has made it possible to increase the reliability, service life and thermal stability of machines of this type. In addition, helical channels for communication between symmetrical working chambers and channels within the rotor for connecting the working chambers to the working-medium inlet and outlet openings are formed inside the spherical rotor part. This solution has made it possible to increase the specific power of the machine.