We expand here the study of the type of identity which has been successfully used to prove the necessity of spherical symmetry in static black holes and uniform density stellar models. We show how the system of partial differential equations, whose solutions correspond to these identities, can be decoupled and partially integrated for fluids with an arbitrary equation of state. The problem of finding such identities is reduced thereby to the problem of finding the solutions to a single ordinary differential equation, plus quadratures.