Strengthening a theorem of D.W. Kueker, this paper completely characterizes which countable structures do not admit uncountable L ω 1 ω -elementarily equivalent models. In particular, it is shown that if the automorphism group of a countable structure M is abelian, or even just solvable, then there is no uncountable model of the Scott sentence of M. These results arise as part of a study of Polish groups with compatible left-invariant complete metrics.