Conformal Killing Initial Data

We find necessary and sufficient conditions ensuring that the vacuum development of an initial data set of the Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID) and they extend the well-known 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 Killing initial data (HKID) conditions than was previously available in the literature.