  In this paper, we mainly consider two class of travelling stratified periodic
water waves, one with negative (or without) surface tension and the other with
constant Bernoulli's function and stagnation points. We first establish the
symmetry result for stratified water waves with negative (or without) surface
tension, but without stagnation by using the modified maximum principle.
Furthermore, the symmetry property of stratified water waves with constant
Bernoulli's function and stagnation points is also obtained provided the
monotonic property is known.
