  We show that the Euclidean Kerr-NUT-(A)dS metric in $2m$ dimensions locally
admits $2^m$ hermitian complex structures. These are derived from the existence
of a non-degenerate closed conformal Killing-Yano tensor with distinct
eigenvalues. More generally, a conformal Killing-Yano tensor, provided its
exterior derivative satisfies a certain condition, algebraically determines
$2^m$ almost complex structures that turn out to be integrable as a consequence
of the conformal Killing-Yano equations. In the complexification, these lead to
$2^m$ maximal isotropic foliations of the manifold and, in Lorentz signature,
these lead to two congruences of null geodesics. These are not shear-free, but
satisfy a weaker condition that also generalizes the shear-free condition from
4-dimensions to higher-dimensions. In odd dimensions, a conformal Killing-Yano
tensor leads to similar integrable distributions in the complexification. We
show that the recently discovered 5-dimensional solution of Lu, Mei and Pope
also admits such integrable distributions, although this does not quite fit
into the story as the obvious associated two-form is not conformal
Killing-Yano. We give conditions on the Weyl curvature tensor imposed by the
existence of a non-degenerate conformal Killing-Yano tensor; these give an
appropriate generalization of the type D condition on a Weyl tensor from four
dimensions.
