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We study unbounded 2-dimensional metric polytopes such as those arising asK\"ahler quotients of complete K\"ahler 4-manifolds with two commutingsymmetries and zero scalar curvature. Under a mild closedness condition, weobtain a complete classification of metrics on such polytopes, and as a resultclassify all possible metrics on on the corresponding K\"ahler 4-manifolds. Ifthe 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 generalizedTaub-NUTs (discovered by Donaldson) are indeed the only possible metrics.Polytopes with $n\ge3$ edges admit an $(n+2)$-dimensional family of possiblemetrics.

Classification of polytope metrics and complete scalar-flat K\"ahler 4-Manifolds with two symmetries