  In this paper, we consider the steadiness of symmetric solutions to two
dispersive models in shallow water and hyperelastic mechanics, respectively.
These models are derived previously in the two-dimensional setting and can be
viewed as the generalization of the Camassa-Holm and Kadomtsev-Petviashvili
equations. For these two models, we prove that symmetry of classical solutions
implies steadiness in the horizontal direction. We also confirm the such
connection between symmetry and steadiness in weak formulation, which includes
in particular the peaked solutions.
