  In this paper, we present a detailed analysis of first-order perturbations of
the Kerr metric in the slow-rotation limit. We perform the calculation by
perturbing the Schwarzschild metric plus up to second-order corrections in the
spin in the Regge-Wheeler gauge. The apparent coupling between different
angular momentum axial-led and polar-led modes can be removed by suitably
combining the perturbation equations and projecting them onto spin-weighted
spherical harmonics. In this way, we derive the corrections to the
Regge-Wheeler and the Zerilli equations up to second-order in the spin. We show
that the two potentials remain isospectral as in the non-rotating limit.
However, it is easy to demonstrate it only for a precise choice of the tortoise
coordinate. The isospectrality with slow-rotating Teukolsky equation is also
verified. We discuss the main implication of this result for the problem of
vacuum metric reconstruction, providing the transformation rule between
slow-spinning Teukolsky variables and metric perturbations. The existence of
this relation leaves us with the conjecture that a resummation of the expansion
in the spin is possible, leading to two decoupled differential equations for
perturbations of the Kerr metric.

General Relativity and Quantum Cosmology