  Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned
null direction (WAND) are analyzed in arbitrary dimension using recently
developed higher-dimensional Newman-Penrose (NP) formalism. We determine
dependence of the metric and of the Weyl tensor on the affine parameter r along
null geodesics generated by the WAND for type III and N spacetimes and for a
special class of type II and D spacetimes, containing e.g.
Schwarzschild-Tangherlini black holes and black strings and branes.
  For types III and N, all metric components are at most quadratic polynomials
in r while for types II and D the r-dependence of the metric as well as of the
Weyl tensor is determined by an integer m corresponding to the rank of the
expansion matrix S_{ij}. It is shown that for non-vanishing expansion, all
these spacetimes contain a curvature singularity.
  As an illustrative example, a shearing expanding type N five-dimensional
vacuum solution is also re-derived using higher-dimensional NP formalism. This
solution can be, however, identified with a direct product of a known
four-dimensional type N metric with an extra dimension.
