  We study unbounded 2-dimensional metric polytopes such as those arising as
K\"ahler quotients of complete K\"ahler 4-manifolds with two commuting
symmetries and zero scalar curvature. Under a mild closedness condition, we
obtain a complete classification of metrics on such polytopes, and as a result
classify all possible metrics on on the corresponding K\"ahler 4-manifolds. If
the polytope is the plane or half-plane then only flat metrics are possible,
and if the polytope has one corner then the 2-parameter family of generalized
Taub-NUTs (discovered by Donaldson) are indeed the only possible metrics.
Polytopes with $n\ge3$ edges admit an $(n+2)$-dimensional family of possible
metrics.
