Integrated Heat Pumps have been defined in the literature, but are only available commercially as custom-built units. Of special promise are integrated heat pumps that are based on the Vuilleumier cycle. They are currently investigated intensively for this application. In this paper a generalized configuration is analyzed theoretically where energy input is in the form of thermal energy only and the system can be used either as a heat pump or a refrigerator. Although Vuilleumier machines are taken as the prime example of an integrated heat pump, the same basic treatment is also applicable to other designs. A new concept of a “Null Engine” is introduced here to establish quantitative relationships between engine proportions and operating characteristics. The generalized heat-operated Vuilleumier machine is analyzed from first principles based on the applicable temperature-entropy diagrams. From this a theoretical volume ratio between the cold and the hot expansion space in a Vuilleumier machine is deduced for the first time. The most important result of this analysis is that a theoretical minimum for operational hardware is established which will ensure an energy balance for an ideal integrated heat pump.