Abstract
We find necessary and sufficient conditions ensuring that the vacuum development of an initial dataset of Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID), and they extend the wellknown Killing initial data (KID) that have been known for a long time.
The procedure used to find the CKID is a classical argument, which is reviewed and presented in a form that may have an independent interest, based on identifying a suitable propagation identity and checking the well-posedness of the corresponding initial value problem. As example applications, we review the derivation of the KID conditions as well as give a more thorough treatment of the homothetic KID conditions than was previously available in the literature.