  We show that horizontally symmetric water waves are traveling waves. The
result is valid for the Euler equations, and is based on a general principle
that applies to a large class of nonlinear partial differential equations,
including some of the most famous model equations for water waves. A detailed
analysis is given for weak solutions of the Camassa-Holm equation. In addition,
we establish the existence of nonsymmetric linear rotational waves for the
Euler equations.
