An air-hybrid engine operates in two operational modes. It employs uniquely designed camshafts with cam lobe activators, and camshaft phase shifters to operate and control the engine valves. It uses a different cam lobe for operating an engine valve during each of the two operational modes, and it uses camshaft phase shifters to vary the valve event timing and duration. The system exploits non-linearity that exists in relationship between the piston motion and the camshaft rotation. Varying the phase relationship between the camshaft and the engine crankshaft in a pre-determined specific way varies the duration of the engine valve opening in the required way, thus varying the volumes of air received into and discharged from the engine.