A rotary pump includes a stator having a chamber delimited by a circumferential wall, a cylindrical rotor tangent to the circumferential wall of the chamber and suitable for being driven in rotation around an axis eccentric with respect to the chamber, and at least a vane diametrically traversing the rotor by delimiting in the rotor two half shells, mutually separated by the walls guiding the vane. The two half shells are hollow, and inside these hollow half shells are housed elastic leaf springs forming one-way valves with respect to four passageways formed in the half shells; the discharge of the air-oil mixture takes place, after a centrifugation and a partial recycling, through passageways, also in the case of a counter rotation of the pump. This configuration allows a bidirectional use of the same rotor.