Abstract: The rotary machine comprises a cylindrical chamber having a rolling piston rotating therein, the piston rolling against the inside surface of the chamber wall. In normal operation, springs keep a vane pressed against the periphery of the rolling piston. A control device for controlling the piston of the vane comprises a control rod lying in the same plane as the vane. The rod includes a retaining abutment formed at the bottom of the control rod and received in a housing formed inside the vane and including a top wall through which the control rod passes, the top wall co-operating with the abutment and the housing having a vertical extent which is not less than the maximum distance between the peripheral surface of the piston and the casing of the rotary machine, thereby enabling the vane to move freely relative to the abutment when the abutment is in its low position.