Based on the decoupling transformation and the Lie point symmetry group method, the (2+1)-D KD equation is reduced to the (1+1)-D nonlinear PDE. By extended homoclinic test approach, new perturbed non-traveling wave double solitary solutions of the (2+1)-D KD equation are obtained. Also, the dynamic critical point and the non-traveling wave rational function singular solutions in the limitation of parameters are derived. Applying the Hamilton function in 2-D planar dynamical system, we discuss the existence of the periodic solutions for the symmetrical and reduced equation with the wave transformation. Moreover, some periodic solutions are derived by the Tan-function test method, and the perturbed non-traveling wave periodic solutions for the (2+1)-D KD equation are shown.
