The Painlevé integrability of a special coupled Jaulent-Miodek equation was studied by using the Kruskal's simplification of WTC method.The following conclusion was obtained:the Jaulent-Miodek equation is of Painlevé integrability in three cases,and is of no Painlevé integrability in one case.Even though in that case,the equation is of no Painlevé integrability,new exact solutions of the coupled Jaulent-Miodek equation can be constructed by using the Painlevé standard and nonstandard truncation expansion.