  A modified-gravity theory with a four-form field strength $F$, a variable
gravitational coupling parameter $G(F)$, and a standard matter action is
considered here. Maxwell and Einstein equations are now derived when including
to action also derivates of $F$. The energy momentum tensor of the 4-form field
contains both the part, which is typical for the fundamental (pseudo)scalar,
and the part, which cancels the divergent contribution of the zero-point
energies of quantum fields to the vacuum energy and thus leads to the natural
nullification of the cosmological constant in Minkowski vacuum.
