First of all, I will explain our variable-splitting theorems (J. Korean Math. Soc. 37 (2000), no. 2, 245--251 and Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 433--448). Then I will formulate a generalization of our theorems as a parameter version. Since I believe that Segre variety mapping is a useful tool to attack the problem, I will give you an another proof of Kim-Zaitsev-A.H. theorem, which appears in above second mentioned paper.