  We present and amplify some of our previous statements on non-canonical
interrelations between the solutions to free Dirac equation (DE) and
Klein-Gordon equation (KGE). We demonstrate that all the solutions to the DE
(possessing point- or string-like singularities) can be obtained via
differentiation of a corresponding pair of the KGE solutions for a doublet of
scalar fields. On this way we obtain a "spinor analogue" of the mesonic Yukawa
potential and previously unknown chains of solutions to DE and KGE, as well as
an exceptional solution to the KGE and DE with a finite value of the field
charge ("localized" de Broglie wave). The pair of scalar "potentials" is
defined up to a gauge transformation under which corresponding solution of the
DE remains invariant. Under transformations of Lorentz group, canonical spinor
transformations form only a subclass of a more general class of transformations
of the solutions to DE upon which the generating scalar potentials undergo
transformations of internal symmetry intermixing their components. Under
continious turn by one complete revolution the transforming solutions, as a
rule, return back to their initial values ("spinor two-valuedness" is absent).
With arbitrary solution of the DE one can associate, apart of the standard one,
a non-canonical set of conserved quantities, positive definite "energy" density
among them, and with any KGE solution -- positive definite "probability
density", etc. Finally, we discuss a generalization of the proposed procedure
to the case when the external electromagnetic field is present
