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We present and enhance our previous statements (arXiv:0907.4736) on thenon-canonical interrelations between the solutions to the free Dirac equation(DE) and the Klein-Gordon equation (KGE). We demonstrate that all the solutionsto the DE (singular ones among them) can be obtained via differentiation of apair of the KGE solutions for a doublet of scalar fields. In this way we obtaina "spinor analogue" of the mesonic Yukawa potential and previously unknownchains of solutions to the DE and KGE. The pair of scalar "potentials" isdefined up to a gauge transformation under which the corresponding solution ofthe DE is invariant. Under transformations of the Lorentz group, canonicalspinor transformations form only a subclass of more general transformations ofthe DE solutions, under these the generating scalar potentials undergotransformations of internal symmetry intermixing their components. Inparticular, under continuous rotation through one full revolution thetransforming solutions, as a rule, take their original values ("spinortwo-valuedness" is absent). With an arbitrary solution of the DE one canassociate, apart from the standard one, a non-canonical set of conservedquantities, a positive definite "energy" density among others, and with any KGEsolution -- a positive definite "probability density", etc. Finally, we discussa generalization of the proposed procedure to the case when an externalelectromagnetic field is present.

Gauge origin of the Dirac field and singular solutions to the Dirac equation