Creating and relating three-dimensional integrable maps

We show how some integrable third-order difference equations recently given in the literature are related to one another by the process of interchanging parameters and integrals. Using the same process, we then create a 21-parameter family of integrable third-order difference equations that contains the previous examples as special cases. Our methodology illustrates that the combination of finding 2-integrals (i.e. integrals of the second iterate of the map), exploiting linear parameter dependence and using the interchange process provides a powerful way to relate and create higher-dimensional discrete integrable systems.