  We provide an explicit resolution of the Abreu equation on convex labeled
quadrilaterals. This confirms a conjecture of Donaldson in this particular case
and implies a complete classification of the explicit toric K\"ahler-Einstein
and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we
obtain a wealth of extremal toric (complex) orbi-surfaces, including
K\"ahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed
points of the torus action, the vanishing of the Futaki invariant is a
necessary and sufficient condition for the existence of K\"ahler metric with
constant scalar curvature. Our results also provide explicit examples of
relative K-unstable toric orbi-surfaces that do not admit extremal metrics.
