Symmetry of steady periodic gravity water waves with vorticity

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Steady Periodic Water Waves with Constant Vorticity: Regularity and Local Bifurcation (2009, unknown): “AMS Mathematics Subject Classification (2000): 76B15, 35J25, 30E25, 35Q35, 35R35.”
Global bifurcation of steady gravity water waves with critical layers (2014, unknown): “[12, 11].”
On Gerstner’s Water Wave (2008, background): “In recent years there has been growing interest in free surface water waves (see [1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 18, 20, 21, 22, 25, 35, 36]), but the current state of affairs regarding explicit solutions has not advanced much since the last century.”
Analyticity of periodic traveling free surface water waves with vorticity (2011, background): “Notice that the symmetry assumptions encompassed in the above definition of a traveling wave solution are not restrictive requirements being actually granted for wave profiles that are monotone between crest and trough [5].”
Pressure beneath a Stokes wave (2009, background): “For a discussion of rotational flows, see [6, 8].”