The past quasi-local horizons in vacuum Robinson-Trautman space-times are described. The case of a null (non-expanding) horizon is discussed. It is shown that the only Robinson-Trautman spacetime admitting such a horizon with sections diffeomorphic to S 2 is the Schwarzschild space-time. Weakening this condition leads to the horizons of the C-metric. Properties of the hypersurface r = 2m are examined.

We develop a formalism that allows us to obtain an approximate Bondi–Sachs form of metrics which are asymptotically flat at null infinity . We find conditions which assure that algebraically special metrics fall into this class. For these metrics we calculate the Bondi mass aspect, the angular momentum aspect and the news function.

We examine conditions assuring that an algebraically special metric is asymptotically flat at future null infinity. Using approximate Bondi-Sachs coordinates we give explicit formulae for the Bondi mass aspect and we interpret one of the Einstein equations as the energy loss formula.

We study point symmetries of the Robinson-Trautman equation. The cases of one-and twodimensional algebras of infinitesimal symmetries are discussed in detail. The corresponding symmetry reductions of the equation are given. Higher dimensional symmetries are shortly discussed. It turns out that all known exact solutions of the Robinson-Trautman equation are symmetric.

We describe conditions assuring that the Kerr-Schild type solutions of Einstein's equations with pure radiation fields are asymptotically flat at future null infinity. Such metrics cannot describe "true" gravitational radiation from bounded sources-it is shown that the Bondi news function vanishes identically. We obtain formulae for the total energy and angular momentum at I + . As an example we consider a non-stationary generalization of the Kerr metric given by Vaidya and Patel. Angular momentum and total energy are expressed in closed form as functions of retarded time.