The gauge freedoms of enlarged Helmholtz theorem and the Neumann ---   Debye potentials; their manifestation in the multipole expansion of conserved   current
  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. 
E.N. Bukina and V.M. Dubovik