In this paper we describe, up to a congruence, translation surfaces in the Galilean space having constant Gaussian and mean curvatures as well as translation Weingarten surfaces. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature K that are not cylindrical surfaces, and translation surfaces with constant mean curvature H 6 0 that are not ruled.
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Šipuš et al. (2011) studied this question.