Abstract: The engine comprises, for each of its cylinders, an auxiliary piston provided with a piston rod. The piston is movable in a cavity which is formed in the cylinder head and opens into the combustion chamber. The cavity defines with the auxiliary piston an annular chamber around the piston rod. A liquid under pressure is supplied to the annular chamber by way of a check-valve. The rod has an axial bore in which is slidable a slide valve member whose position is under the control of a parameter representing the conditions of operation of the engine. The slide valve member cooperates with at least one escape port formed across the hollow rod to constitute a liquid discharge path for the annular chamber.