  In this paper we discuss Airy solutions of the second Painlev\'e equation
(\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlev\'e XXXIV
equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of
\mbox{\rm P$_{\rm II}$}\ (\mbox{\rm S$_{\rm II}$}), are discussed. It is shown
that solutions which depend only on the Airy function $\mathop{\rm
Ai}\nolimits(z)$ have a completely difference structure to those which involve
a linear combination of the Airy functions $\mathop{\rm Ai}\nolimits(z)$ and
$\mathop{\rm Bi}\nolimits(z)$. For all three equations, the special solutions
which depend only on $\mathop{\rm Ai}\nolimits(t)$ are \textit{tronqu\'ee}
solutions, i.e.\ they have no poles in a sector of the complex plane. Further
for both $\mbox{\rm P}_{34}$\ and \mbox{\rm S$_{\rm II}$}, it is shown that
amongst these \textit{tronqu\'ee} solutions there is a family of solutions
which have no poles on the real axis.
