  Abreu-Sena-Dias have constructed two distinct families of scalar-flat
K\"ahler non-compact toric metrics using Donaldson's rephrasing of Joyce's
construction in action-angle coordinates. In this paper and using the same
set-up, we show that these are the only J-complete scalar-flat K\"ahler metrics
on any given strictly unbounded toric surface. We also show that the asymptotic
behaviour of such a metric determines it uniquely.
