  This work presents the main aspects of the anisotropic gravity-vector gauge
coupling at all energy scales \i.e., from the IR to the UV point. This study is
carry out starting from the 4+1 dimensional Ho\v{r}ava-Lifshitz theory, at the
kinetic conformal point.The Kaluza-Klein technology is employed as a unifying
mechanism to couple both interactions. Furthermore, by assuming the so-called
cylindrical condition, the dimensional reduction to 3+1 dimensions leads to a
theory whose underlying group of symmetries corresponds to the diffeomorphisms
preserving the foliation of the manifold and a U(1) gauge symmetry. The
counting of the degrees of freedom shows that the theory propagates the same
spectrum of Einstein-Maxwell theory. The speed of propagation of tensorial and
gauge modes is the same, in agreement with recent observations. This point is
thoroughly studied taking into account all the $z=1,2,3,4$ terms that
contribute to the action. In contrast with the 3+1 dimensional formulation,
here the Weyl tensor contributes in a non-trivial way to the potential of the
theory. Its complete contribution to the 3+1 theory is explicitly obtained.
Additionally, it is shown that the constraints and equations determining the
full set of Lagrange multipliers are elliptic partial differential equations of
eighth-order. To check and assure the consistency and positivity of the reduced
Hamiltonian some restrictions are imposed on the coupling constants. The
propagator of the gravitational and gauge sectors are obtained showing that
there are not ghost fields, what is more they exhibit the $z=4$ scaling for all
physical modes at the high energy level. By evaluating the superficial degree
of divergence and considering the structure of the second class constraints, it
is shown that the theory is power-counting renormalizable.
