“…By introducing hodograph transformation, the result was further improved in [5] for any arbitrary vorticity functions. Following the breakthrough work in [5], the symmetry properties of periodic water waves have been extended to more general rotational flows, such as flows are stratified [3,6], or with negative surface tension [19] and flows are near equatorial region [20,29], or with arbitrarily many crests and troughs [28]. Besides, the moving plane method was also applied to investigate the symmetry properties of gravity solitary waves in [22,23].…”