  It is shown that a self-dual neutral Einstein four-manifold of Petrov type
III, admitting a two-dimensional null parallel distribution compatible with the
orientation, cannot be compact or locally homogeneous, and its maximum possible
degree of mobility is 3. Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo found a
general coordinate form of such manifolds. The present paper also provides a
modified version of that coordinate form, valid in a suitably defined generic
case and, in a sense, "more canonical" than the usual formulation. Moreover,
the local-isometry types of manifolds as above having the degree of mobility
equal to 3 are classified. Further results consist in explicit descriptions,
first, of the kernel and image of the Killing operator for any torsionfree
surface connection with everywhere-nonzero, skew-symmetric Ricci tensor, and,
secondly, of a moduli curve for surface connections with the properties just
mentioned that are, in addition, locally homogeneous. Finally, hyperbolic plane
geometry is used to exhibit examples of codimension-two foliations on compact
manifolds of dimensions greater than 2 admitting a transversal torsionfree
connection, the Ricci tensor of which is skew-symmetric and nonzero everywhere.
No such connection exists on any closed surface, so that there are no analogous
examples in dimension 2.
