  It has recently been shown that the dynamics of perturbed non-rotating black
holes (BHs) admits an infinite number of symmetries that are generated by the
flow of the Korteweg-de Vries (KdV) equation. These symmetries lead to an
infinite number of conserved quantities that can be obtained as integrals of
differential polynomials in the potential appearing in the gauge-invariant
master equations describing the BH perturbations, the KdV integrals. These
conserved quantities are the same for all the possible potentials, which means
that they are invariant under Darboux transformations, and they fully determine
the BHs transmission amplitudes, or greybody factors, via a moment problem. In
this paper we introduce a new semi-analytical method to obtain the greybody
factors associated with BH scattering processes by solving the moment problem
using only the KdV integrals. The method is based on the use of Pad\'e
approximants and we check it first by comparing with results from the case of a
P\"oschl-Teller potential, for which we have analytical expressions for the
greybody factors. Then, we apply it to the case of a Schwarzschild BH and
compare with results from computations based on the Wentzel-Kramers-Brillouin
(WKB) approximation. It turns out that the new method provides accurate results
for the BH greybody factors for all frequencies. The method is also
computationally very efficient.
