  We present and enhance our previous statements (arXiv:0907.4736) on the
non-canonical interrelations between the solutions to the free Dirac equation
(DE) and the Klein-Gordon equation (KGE). We demonstrate that all the solutions
to the DE (singular ones among them) can be obtained via differentiation of a
pair of the KGE solutions for a doublet of scalar fields. In this way we obtain
a "spinor analogue" of the mesonic Yukawa potential and previously unknown
chains of solutions to the DE and KGE. The pair of scalar "potentials" is
defined up to a gauge transformation under which the corresponding solution of
the DE is invariant. Under transformations of the Lorentz group, canonical
spinor transformations form only a subclass of more general transformations of
the DE solutions, under these the generating scalar potentials undergo
transformations of internal symmetry intermixing their components. In
particular, under continuous rotation through one full revolution the
transforming solutions, as a rule, take their original values ("spinor
two-valuedness" is absent). With an arbitrary solution of the DE one can
associate, apart from the standard one, a non-canonical set of conserved
quantities, a positive definite "energy" density among others, and with any KGE
solution -- a positive definite "probability density", etc. Finally, we discuss
a generalization of the proposed procedure to the case when an external
electromagnetic field is present.
