The authors prove that is the only solution of the system
if , , and but not both equal to , which answers in the affirmative (a generalization of) a conjecture by J. Serrin and H. Zou [Discourses Math. Appl. 3, 55–68 (1994; Zbl 0900.35121)]. The technique used is a moving plane method applied to Kelvin-type transforms of and .