The Painlevé analysis method developed by Weiss,Tabor and Carnevale is one of the most effective and extensively used methods to test the integrability of the nonlinear partial differential equation.With the help of symbolic computation system Maple,the (2+1)-dimensional Lax-Kadomtsev-Patviashvili (Lax-KP) equation was proved to be Painlevé non-integrable by using the Weiss,Tabor and Carnelvale (WTC) method.The leading order analysis helps one to find two cases and verify that the recursion relations are established directly.New exact solutions of the (2+1)-dimensional Lax-KP equation were obtained by the standard and nonstandard truncation expansions respectively,and all the solutions are both kink solitary solutions when selecting proper constants.