  We discuss the existence of asymptotically Euclidean initial data sets to the
vacuum Einstein field equations which would give rise (modulo an existence
result for the evolution equations near spatial infinity) to developments with
a past and a future null infinity of different smoothness. For simplicity, the
analysis is restricted to the class of conformally flat, axially symmetric
initial data sets. It is shown how the free parameters in the second
fundamental form of the data can be used to satisfy certain obstructions to the
smoothness of null infinity. The resulting initial data sets could be
interpreted as those of some sort of (non-linearly) distorted Schwarzschild
black hole. Its developments would be so that they admit a peeling future null
infinity, but at the same time have a polyhomogeneous (non-peeling) past null
infinity.
