  Extended $U_e(1)\times U_g(1)$ electromagnetism containing both a photon and
a pseudo-photon is introduced at variational level and is justified by the
violation of the Bianchi identities in conceptual systems, either in the
presence of magnetic monopoles or non-regular external fields, not being
accounted by the standard Maxwell Lagrangian. It is carried a dimensional
reduction that renders a $U_e(1)\times U_g(1)$ Maxwell-BF type theory and it is
considered canonical functional quantization in planar systems which may be
relevant in Hall systems.
