  We consider the propagation of smooth solitary waves in a two-dimensional
generalization of the Camassa--Holm equation. We show that transverse
perturbations to one-dimensional solitary waves behave similarly to the KP-II
theory. This conclusion follows from our two main results: (i) the double
eigenvalue of the linearized equations related to the translational symmetry
breaks under a transverse perturbation into a pair of the asymptotically stable
resonances and (ii) small-amplitude solitary waves are linearly stable with
respect to transverse perturbations.
