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