  Complex and real, vacuum spaces with both self-dual and anti-self-dual parts
of the Weyl tensor being of the type [N] are considered. Such spaces are
classified according to two criteria. The first one takes into account the
properties of the congruences of totally null, geodesic 2-dimensional surfaces
(the null strings). The second criterion is based on investigations of the
properties of the intersection of these congruences. It is proved that there
exist six distinct types of the $[\textrm{N}] \otimes [\textrm{N}]$ spaces. New
examples of the Lorentzian slices of the complex metrics are presented. Also,
some type $[\textrm{N}] \otimes [\textrm{N}]$ spaces which do not posses
Lorentzian slices are considered.
