  We study surfaces in TN that are area-stationary with respect to a neutral
Kaehler metric constructed on TN from a riemannian metric g on N. We show that
holomorphic curves in TN are area-stationary, while lagrangian surfaces that
are area-stationary are also holomorphic and hence totally null. However, in
general, area stationary surfaces are not holomorphic. We prove this by
constructing counter-examples. In the case where g is rotationally symmetric,
we find all area stationary surfaces that arise as graphs of sections of the
bundle TN$\to$N and that are rotationally symmetric. When (N,g) is the round
2-sphere, TN can be identified with the space of oriented affine lines in
${\Bbb{R}}^3$, and we exhibit a two parameter family of area-stationary tori
that are neither holomorphic nor lagrangian.
