  We consider the existence of Taub-NUT solutions in third order Lovelock
gravity with cosmological constant, and obtain the general form of these
solutions in eight dimensions. We find that, as in the case of Gauss-Bonnet
gravity and in contrast with the Taub-NUT solutions of Einstein gravity, the
metric function depends on the specific form of the base factors on which one
constructs the circle fibration. Thus, one may say that the independence of the
NUT solutions on the geometry of the base space is not a robust feature of all
generally covariant theories of gravity and is peculiar to Einstein gravity. We
find that when Einstein gravity admits non-extremal NUT solutions with no
curvature singularity at $r=N$, then there exists a non-extremal NUT solution
in third order Lovelock gravity. In 8-dimensional spacetime, this happens when
the metric of the base space is chosen to be $\Bbb{CP}^{3}$. Indeed, third
order Lovelock gravity does not admit non-extreme NUT solutions with any other
base space. This is another property which is peculiar to Einstein gravity. We
also find that the third order Lovelock gravity admits extremal NUT solution
when the base space is $T^{2}\times T^{2}\times T^{2}$ or $S^{2}\times
T^{2}\times T^{2}$. We have extended these observations to two conjectures
about the existence of NUT solutions in Lovelock gravity in any
even-dimensional spacetime.
