  Attention is drawn to the mathematical equality of rights of symmetrical
constituents derived affinor of a vector field in relation to its antisymmetric
constituents. In this regard, raises the question not only of equitable
accounting, but and mainly question of the real existence of fields,
represented by these constituents. In particular, we conclude that the
classical electromagnetic field at any point of space-time accompanied, in the
General case, independent physical field, defined symmetrical derived affinor
of 4-potential of classical electrodynamics. Discussed, within the framework of
the Bogolyubov and Shirkov axiomatic, a theory of real vector field, clearly
and equitably taking into account the symmetric derived affinors this field and
found a number of important distinguishing features this model. Despite
accounting explicitly gauge-noninvariant constituents, the proposed theory has
specialized gauge invariance, which provides, in particular, conservation of
electric current. In this connection, the difficulties with the probabilistic
interpretation, in the case of indefinite-metric version of the theory, are
surmountable by standard methods. Finally, is proposed the physical content of
the fields, defined of a symmetrical derived affinor of 4-potential of
classical electrodynamics.
  Keywords: vector field, symmetrical derived affinor, gauge invariance.
