A rotary displacement machine includes a casing having a circular cylindrical internal surface and a rotor disposed in the chamber mounted to orbit about a chamber axis, with the rotor having a circular cylindrical external surface. A vane member is mounted on the casing and pivotable about a pivot axis parallel to the chamber axis. The vane member has a passage way communicating between the exterior of the casing and the operating chamber. A linkage connects the vane member to the rotor so as to keep a tip face of the vane member in sealing contact with an external surface of the rotor.