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In this paper, we investigate a nonlinear evolution equation which is derived from plane curve by Lie symmetry analysis and theory of planar dynamical system. The one-dimensional optimal system of the equation is given and all bifurcations of the system in different parametric regions are given from which we can guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions and many uncountable infinite traveling wave solutions. Besides, we provide exact expressions of traveling wave solutions under different parametric conditions.