A positive displacement pump comprising: - a casing that comprises a central body and two closing lids , said central body being provided with two communicating cylindrical chambers , one suction pipe (I) and one discharge pipe (O), and - two rotors revolvingly mounted in said chambers of the central body and supported by corresponding shafts revolvingly mounted in said closing lids , said two rotors comprise: - a male rotor that only comprises at least one protuberance , not cavities, and - a female rotor that only comprises at least one cavities , not protuberances or teeth, wherein the male rotor mesh with the female rotor , i.e. the at least one protuberance of the male rotor is engaged in the at least one cavities of the female rotor without contact between the two rotors, and the male rotor comprises a body provided with at least one seat and said at least one protuberance consist in a sector comprising a base that is engaged in the seat of the body of the rotor. wherein said male rotor comprises a cylindrical body and at least one protuberance that protrude radially from the cylindrical body and said female rotor comprises a cylindrical body and at least one cavity that extend radially inside the cylindrical body, whereby - each protuberance of the male rotor cross-sectionally comprises two flex-shaped sides converging into a rounded or flat head , the two sides of a protuberance being symmetrical with respect to a radial axis of symmetry passing through the head of the protuberance, and - each cavity cross-sectionally comprises two flex-shaped sides comprising an inlet side and an outlet side having flex shaped profiles joined into a bottom surface with concave shape, wherein the flex shaped side of the protuberance and cavity is a curve that has a flex; the flex or inflection is a point on a curve at which the curvature or concavity changes sign from plus to minus or from minus to plus; the curve changes from being concave to convex, or vice versa; characterized in that the profiles of the two sides of the cavity are not symmetrical with respect to a radial straight line passing through the bottom of the cavity, the flex-shaped profile of the inlet side having a higher curvature than the flex-shaped profile of the outlet side of the cavity; wherein the female rotor and the male rotor are disposed side by side and the suction and delivery ducts (I, O) are disposed on one side and on the other side with respect to the rotors and the suction and delivery ducts (I, O) have axes orthogonal each others. - a casing that comprises a central body and two closing lids , said central body being provided with two communicating cylindrical chambers , one suction pipe (I) and one discharge pipe (O), and - two rotors revolvingly mounted in said chambers of the central body and supported by corresponding shafts revolvingly mounted in said closing lids , - a male rotor that only comprises at least one protuberance , not cavities, and - a female rotor that only comprises at least one cavities , not protuberances or teeth, - each protuberance of the male rotor cross-sectionally comprises two flex-shaped sides converging into a rounded or flat head , the two sides of a protuberance being symmetrical with respect to a radial axis of symmetry passing through the head of the protuberance, and - each cavity cross-sectionally comprises two flex-shaped sides comprising an inlet side and an outlet side having flex shaped profiles joined into a bottom surface with concave shape, <EOS>