  Based on the covariant variation formalism, two versions of the symmetric
effective stress-energy tensor of the electromagnetic field in a
dynamo-optically active relativistic media are reconstructed in the framework
of the tetrad and aether paradigms, respectively. We show that the energy
density scalars and pressure tensors coincide for both versions of the
stress-energy tensors, however, the corresponding energy flux four-vectors
happen to be different in general case. This mathematical fact adds new
arguments into the 100-year-long discussion, which is called Minkowski-Abraham
controversy and is connected with the correct definition of the electromagnetic
energy flux in a continuous media. We consider three examples: first, the
axionically active vacuum; second, the spatially isotropic moving dielectric
medium; third, the dynamo-optically active medium. We discuss possible
applications of the elaborated formalism.
