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We study properties of the Maxwell electromagnetic invariant in the externalregion of spinning and charged horizonless stars. We analytically find that theminimum negative value of the Maxwell electromagnetic invariant is obtained onthe equator of the star surface. We are interested in scalar fieldsnon-minimally coupled to the Maxwell electromagnetic invariant. The negativeenough Maxwell electromagnetic invariant can lead to a negative effective massterm, which forms a binding potential well for the scalar field. It means thatthe scalar field coupled to the Maxwell electromagnetic invariant may mostlyexist around the surface of the star on the equator.

Analytical investigations on the Maxwell electromagnetic invariant in the spinning and charged horizonless star background