  We prove a one-dimensional symmetry result for a weighted
Dirichlet-to-Neumann problem arising in a model for water waves in dimension 3.
More precisely we prove that minimizers and bounded monotone solutions depend
on only one Euclidean variable. The analogue of this result for the
2-dimensional case (and without weights) was established in an article by De La
Llave and the third author. In this paper, a crucial ingredient in the proof is
given by an energy estimate for minimizers obtained via a comparison argument.
