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 that have been known for a long time. The procedure used to find the CKID is a classical argument based on the computation of a suitable propagation identity. The propagation identity is valid in any pseudo-Riemannian signature. In Lorentzian signature it involves hyperbolic operators, while in Riemannian it involves elliptic ones, which may be of independent interest. (Based on arXiv:1905.01231, with A. García-Parrado.)