A balanced rotary cycling machine suitable for use as an internal combustion engine, compressed gas or steam engine, compressor or pump is disclosed herein. The rotor assembly consists of four articulating pistons where the opposite pistons are inter-linked with each other by pivoted rods comprising a parallelogram mechanism and therefore eliminating a need for pivots between pistons. The rotor assembly rotates inside or outside of a circular or non-circular stator depending on the configuration chosen. A variety of mechanisms for shape deformation of four piston assembly during its rotating cycle is also disclosed herein, as well as detailed descriptions of preferred embodiments, including a four cycle internal combustion engine with circular stator, marine engine with polymer parts and a four cycle automobile rotary engine with conventional oil pan. In addition, a method of operation of external rotary combustion engine, employing a high-pressure compressor and an external combustion chamber, is disclosed. This invention also teaches a novel lubrication system for rotary engine providing low emissions. The engine has few moving parts, simplified circular or semi-circular stator shape and utilizes simple and effective sealing techniques. It is fully balanced, has very low friction and heat losses and can employ a pre-designed configuration of the combustion chamber for a desired compression ratio and power output.