  Finsler geometry is a natural generalization of pseudo-Riemannian geometry.
It can be motivated e.g. by a modified version of the Ehlers-Pirani-Schild
axiomatic approach to space-time theory. Also, some scenarios of quantum
gravity suggest a modified dispersion relation which could be phrased in terms
of Finsler geometry. On a Finslerian spacetime, the Universality of Free Fall
is still satisfied but Local Lorentz Invariance is violated in a way not
covered by standard Lorentz Invariance Violation schemes. In this paper we
consider a Finslerian modification of Maxwell's equations. The corrections to
the Coulomb potential and to the hydrogen energy levels are computed. We find
that the Finsler metric corrections yield a splitting of the energy levels.
Experimental data provide bounds for the Finsler parameters.
