  We study N=2(4) superstring backgrounds which are four-dimensional
non-\Kahlerian with non-trivial dilaton and torsion fields. In particular we
consider the case that the backgrounds possess at least one $U(1)$ isometry and
are characterized by the continual Toda equation and the Laplace equation. We
obtain a string background associated with a non-trivial solution of the
continual Toda equation, which is mapped, under the T-duality transformation,
to the hyper-\Kahler Taub-NUT instanton background. It is shown that the
integrable property of the non-\Kahlerian spaces have the direct origin in the
real heavens: real, self-dual, euclidean, Einstein spaces. The Laplace equation
and the continual Toda equation imposed on quasi-\Kahler geometry for
consistent string propagation are related to the self-duality conditions of the
real heavens with ``translational'' and ``rotational''Killing symmetry
respectively.
