  We study properties of the Maxwell electromagnetic invariant in the external
region of spinning and charged horizonless stars. We analytically find that the
minimum negative value of the Maxwell electromagnetic invariant is obtained on
the equator of the star surface. We are interested in scalar fields
non-minimally coupled to the Maxwell electromagnetic invariant. The negative
enough Maxwell electromagnetic invariant can lead to a negative effective mass
term, which forms a binding potential well for the scalar field. It means that
the scalar field coupled to the Maxwell electromagnetic invariant may mostly
exist around the surface of the star on the equator.
