  From the field equations in the linear regime of the characteristic
formulation of general relativity, Bishop, for a Schwarzschild's background,
and M\"adler, for a Minkowski's background, were able to show that it is
possible to derive a fourth order ordinary differential equation, called master
equation, for the $J$ metric variable of the Bondi-Sachs metric. Once $\beta$,
another Bondi-Sachs potential, is obtained from the field equations, and $J$ is
obtained from the master equation, the other metric variables are solved
integrating directly the rest of the field equations. In the past, the master
equation was solved for the first multipolar terms, for both the Minkowski's
and Schwarzschild's backgrounds. Also, M\"adler recently reported a
generalisation of the exact solutions to the linearised field equations when a
Minkowski's background is considered, expressing the master equation family of
solutions for the vacuum in terms of Bessel's functions of the first and the
second kind. Here, we report new solutions to the master equation for any
multipolar moment $l$, with and without matter sources in terms only of the
first kind Bessel's functions for the Minkowski, and in terms of the Confluent
Heun's functions (Generalised Hypergeometric) for radiative (nonradiative) case
in the Schwarzschild's background. We particularize our families of solutions
for the known cases for $l=2$ reported previously in the literature and find
complete agreement, showing the robustness of our results.
