The present invention relates to a mechanism (1) having rotary pistons (21), said mechanism comprising an outer enclosure (2) forming a stator in which a rotary assembly (20) forming a rotor moves, the rotary assembly (20) including: a plurality of pistons (21) forming a hinged polygon, each piston (21) defining a positive-displacement chamber (35) with the enclosure (2), and first guide (27, 27a, 27b) that are arranged so as to engage with second guide means (7a, 7b) comprising rolling surfaces (10a, 10b) so as to drive the polygon to carry out a predetermined movement, the mechanism (1) being characterized in that it comprises a drawback device (50) that maintains contact between the guide means (27, 27a, 27b) and the rolling surfaces (10a, 10b). The invention also relates to an engine and a pump including such a mechanism (1).