We establish the crest-line symmetry of two-dimensional stratified solitary waves under non-stagnation conditions, where the fluid density varies with the stream function and the free surface remains monotonic on each side of the wave crest. By exploiting the elliptic structure of the governing equations, we rigorously prove that: 1. The maximum principle holds for the transformed unbounded boundary value problem. 2. All solutions satisfy reflection symmetry across the vertical crest line. Our proof adapts the moving plane method to stratified solitary waves, overcoming two significant difficulties: The coupling between density and the stream function in the vorticity equation; The unbounded domain with an unknown free boundary.
Feng et al. (Fri,) studied this question.