  A generalized Weyl integrable geometry (GWIG) is obtained from simultaneous
affine transformations of the tangent and cotangent bundles of a
(pseudo)-Riemannian manifold. In comparison with the classical Weyl integrable
geometry (CWIG), there are two generalizations here: interactions with an
arbitrary dark field, and, anisotropic dilation. It means that CWIG already has
interactions with a {\it null} dark field. Some classical mathematics and
physics problems may be addressed in GWIG. For example, by derivation of
Maxwell's equations and its sub-sets, the conservation, hyperbolic, and
elliptic equations on GWIG; we imposed interactions with arbitrary dark fields.
Moreover, by using a notion analogous to Penrose conformal infinity, one can
impose boundary conditions canonically on these equations. As a prime example,
we did it for the elliptic equation, where we obtained a singularity-free
potential theory. Then we used this potential theory in the construction of a
non-singular model for a point charged particle. It solves the difficulty of
infinite energy of the classical vacuum state.
