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In [Abreu and Sena‐Dias,  Ann. Global Anal. Geom . 41 (2012) 209–239], the authors construct two distinct families of scalar‐flat Kähler 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 scalar‐flat Kähler metrics on any given strictly unbounded toric surface. We also show that the asymptotic behaviour of such a metric determines it uniquely if the metric satisfies some extra mild assumptions.

Uniqueness among scalar‐flat Kähler metrics on non‐compact toric 4‐manifolds