[en] In two previous papers we expressed the total energy of the spin multiplicities of a multi-fermionic system, within an orbit induced by a local-scale transformation with scalar function f(r), as an exact functional of the components of the single-particle density p(r). In this paper we provide an integral-differential equation implicit in p(r) and show how to solve it interactively using f(r). We also show how to derive a similar equation explicit in f(r) after expressing the total energy as a functional of f. (authors). 12 refs