In Vaidya spacetime, for linear time -depended mass function, there is a conformal Killing vector. One can do coordinate transformation and this conformal Killing vector becomes (1,0,0,0). This Killing vector generates a constant of motion (energy) along null geodesics. My question is: do conformal Killing vectors generate constants of motion along time-like geodesics? Or it is true only in null case? Also, can anyone tell me please about papers or textbooks where i can learn it deeper?