  We discuss gauge freedom within the scope of the enlarged Helmholtz theorem
and Neumann-Debye decomposition and then demonstrate its realization for the
multipole expansion of a electromagnetic current with distinguished toroid
moment family. The exact solution to the latter problem was obtained in 1974,
but answers to some purely mathematical questions raised by I.B.~French and
Y.~Shimamoto~\cite{French} about 40 years ago are given in this paper for the
first time.
