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This paper aims to show that the Dirac equation coupled to an arbitraryinhomogeneous flux field admits separation in manifolds formed from the directproduct of bidimensional spaces. As a direct application of these results, westudy a spin-$1/2$ charged particle propagating in a background conformallyrelated to a novel and complex string-inspired model in $D = 10$ spacetimedimension whose base manifold is a product of four two-dimensional unitspheres, $S^{2}$, in the presence of $1$- and $3$-form fluxes.

Integrability of the Dirac Equation in the Presence of Fluxes on Product Manifolds