Abstract: Catalyst is supported at least on a part of the combustion chamber wall of an internal combustion engine. A control is provided for controlling the fuel injection equipment of the engine so that pilot fuel injection is performed prior to main fuel injection under prescribed operating conditions of the engine. A control is provided for controlling so that the exhaust valve is reopened for a prescribed period at prescribed timing, or the inlet valve is reopened for a prescribed period at prescribed timing, under prescribed operating conditions of the engine. By the combination of providing catalyst on the combustion wall and controlling fuel injection and valve overlapping, the exhaust emission of internal combustion engine can be drastically improved. The catalyst supporting structure to carry catalyst on the heat insulating material coated on a wall to which combustion gas contacts can be effectively applied to every type of heat engine.