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Attention is drawn to the mathematical equality of rights of symmetricalconstituents derived affinor of a vector field in relation to its antisymmetricconstituents. In this regard, raises the question not only of equitableaccounting, but and mainly question of the real existence of fields,represented by these constituents. In particular, we conclude that theclassical electromagnetic field at any point of space-time accompanied, in theGeneral case, independent physical field, defined symmetrical derived affinorof 4-potential of classical electrodynamics. Discussed, within the framework ofthe Bogolyubov and Shirkov axiomatic, a theory of real vector field, clearlyand equitably taking into account the symmetric derived affinors this field andfound a number of important distinguishing features this model. Despiteaccounting explicitly gauge-noninvariant constituents, the proposed theory hasspecialized gauge invariance, which provides, in particular, conservation ofelectric current. In this connection, the difficulties with the probabilisticinterpretation, in the case of indefinite-metric version of the theory, aresurmountable by standard methods. Finally, is proposed the physical content ofthe fields, defined of a symmetrical derived affinor of 4-potential ofclassical electrodynamics.  Keywords: vector field, symmetrical derived affinor, gauge invariance.

On the theory of vector field with a symmetric affinors. Real vector field in the framework of the standard methods