In this paper, we consider the bifurcations and traveling wave solutions for the generalized \(A\pm \) equation, which we construct from the Alexeyev’s \(A\pm \) equation. Based on the theory of dynamical system, we study the corresponding traveling wave system and bifurcations of its phase orbit portraits. Furthermore, according to the energy level curves of the phase portraits, we analyze all kinds of bounded solutions, and derive the exact representations for the solutions including periodic peakons, peakons, smooth periodic wave solutions, solitary wave solutions, kink wave solutions as well as compacton solution family.
