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In this paper, we discuss Airy solutions of the second Painlevé equation (P II ) and two related equations, the Painlevé XXXIV equation (   P  34   ) and the Jimbo–Miwa–Okamoto σ form of P II  (S II ), are discussed. It is shown that solutions that depend only on the Airy functionAi   (  z  )    have a completely different structure to those that involve a linear combination of the Airy functionsAi   (  z  )    andBi   (  z  )   . For all three equations, the special solutions that depend only onAi   (  t  )    are  tronquée  solutions, i.e., they have no poles in a sector of the complex plane. Further, for both    P  34    and S II , it is shown that among these  tronquée  solutions there is a family of solutions that have no poles on the real axis.

On Airy Solutions of the Second Painlevé Equation