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We classify all scalar-flat toric K\"ahler 4-manifolds under either of twoasymptotic conditions: that the action fields decay slowly (or at all), or thatthe curvature decay is quadratic; for example we fully classify instantons thathave any of the ALE-F-G-H asymptotic types. The momentum functions satisfy adegenerate elliptic equation, and under either asymptotic condition the imageof the moment map is closed. Using a recent Liouville theorem fordegenerate-elliptic equations, we classify all possibilities for the momentumfunctions, and from this, all possible metrics.

A classification of scalar-flat toric K\"ahler instantons in dimension 4