  We derive a system of equations governing the coupled gravitational and
electromagnetic perturbations of Reissner-Nordstr\"om spacetime. The equations
are derived in the context of global non-linear stability of
Reissner-Nordstr\"om under axially symmetric polarized perturbations, as a
generalization of the recent work on non-linear stability of Schwarzschild
spacetime of Klainerman-Szeftel. The main result consists in deriving, through
a Chandrasekhar-type transformation, a gauge invariant quantity associated to
the electromagnetic tensor that verifies a Regge-Wheeler equation. In this
paper, we present the derivation of the main equations, postponing the
estimates of the non-linear system and the construction of the dynamical
spacetime to a later time.
