In this (very good) notes: http://people.physics.tamu.edu/pope/geomlec.pdf it is set as an exercise to proof that if $u_i$ solves the Klein-Gordon equation:
$$(\Box -m^2 )u_i = 0$$
then you can proof, appealing to the properties of the Killing vectors, that the Killing vector $K^\mu \partial_\mu$ also solves
$$(\Box -m^2 )K^\mu \partial_\mu u_i = 0$$
and that the key point is to prove that $\Box(K^\mu \partial_\mu u_i ) = K^\mu \partial_\mu\Box u_i$
Honestly, I don't get with the key. Any help will be appreciated.