A note on the existence of standard splittings for conformally stationary spacetimes

Let(M, g) be a spacetime which admits a complete timelike conformal Killing vector field K. We prove that (M, g) splits globally as a standard conformastationary spacetime with respect to K if and only if (M, g) is distinguishing (and, thus causally continuous). Causal but non-distinguishing spacetimes with complete stationary vector fields are also exhibited. For the proof, the recently solved 'folk problems' on smoothability of time functions (moreover, the existence of a temporal function) are used.