A pumping machine, that can serve a system as the sole main pump for pressurizing a primary liquid flow, incorporates, in a single machine, a rotor-drum type AP (axial piston) pump and a PX (pressure exchanger) that recovers energy from a secondary liquid flow such as the brine discharge from an RO seawater desalination system, with benefits including fewer moving parts and small machine size along with lower capital and operating costs. A single rotor-drum containing the cylinders and pistons is located between two end blocks, one or both configured with manifold passageways, ports and sliding valves. A swash-plate at one end reciprocates the pistons axially when the rotor-drum is rotated. Two working chambers, primary and secondary, are formed at opposite ends of a single piston in each cylinder, thus enabling the single rotor-drum to function as a primary liquid-pressurizing axial pump (AP) with sliding valves at the primary end enabling primary liquid pumping, and as a secondary outflow-driven pressure exchanger (PX) recovering energy from pressure drop in the secondary liquid flow and thus contributing work to primary pumping, saving energy and reducing operating costs.