We announce a generalization of the reduction theorem for 0-parameter solutions of the traditional (i.e., second order) Painlevé equations with a large parameter to those of some higher order Painlevé equations, i.e., each member of the Painlevé hierarchies $(P_J)$ ($J = \mbox{I}$, II-1 and II-2) discussed in [KKNT]. Thus the scope of applicability of the reduction theorem ([KT1, KT2]) has been substantially enlarged; only six equations were covered by our previous result, while the result reported here applies to infinitely many equations.