  We study the totally null surfaces of the neutral Kaehler metric on certain
4-manifolds. The tangent spaces of totally null surfaces are either self-dual
($\alpha$-planes) or anti-self-dual ($\beta$-planes) and so we consider
$\alpha$-surfaces and $\beta$-surfaces. The metric of the examples we study,
which include the spaces of oriented geodesics of 3-manifolds of constant
curvature, are anti-self-dual, and so it is well-known that the $\alpha$-planes
are integrable and $\alpha$-surfaces exist. These are holomorphic Lagrangian
surfaces, which for the geodesic spaces correspond to totally umbilic
foliations of the underlying 3-manifold.
  The $\beta$-surfaces are less known and our interest is mainly in their
description. In particular, we classify the $\beta$-surfaces of the neutral
Kaehler metric on $TN$, the tangent bundle to a Riemannian 2-manifold $N$.
These include the spaces of oriented geodesics in Euclidean and Lorentz
3-space, for which we show that the $\beta$-surfaces are affine tangent bundles
to curves of constant geodesic curvature on $S^2$ and $H^2$, respectively. In
addition, we construct the $\beta$-surfaces of the space of oriented geodesics
of hyperbolic 3-space.
