  New type III and type N approximate solutions which are regular in the linear
approximation are shown to exist. For that, we use complex transformations on
self-dual Robinson-Trautman metrics rather then the classical approach. The
regularity criterion is the boundedness and vanishing at infinity of a scalar
obtained by saturating the Bel-Robinson tensor of the first approximation by a
time-like vector which is constant with respect to the zeroth approximation.
