  A system of equations that serves as a model for the Einstein field equation
in generalised harmonic gauge called the good-bad-ugly system is studied in the
region close to null and spatial infinity in Minkowski spacetime. This analysis
is performed using H. Friedrich's cylinder construction at spatial infinity and
defining suitable conformally rescaled fields. The results are translated to
the physical set up to investigate the relation between the polyhomogeneous
expansions arising from the analysis of linear fields using the $i^0$-cylinder
framework and those obtained through a heuristic method based on H\"ormander's
asymptotic system.
