1. Hirota, R.:Phys. Rev. Lett. 27 (1971), 1192.
2. Hirota, R.: in: R. K. Bullough and P. J. Caudrey (Eds),Soliton, Springer (1980), p. 157.
3. Hietarinta, J.: Hirota's bilinear method and partial integrability, NATO ASI C3 10 (1990), p. 459.
4. This is an oversimplified statement that does not accurately reproduce Painlevé's argument. We refer the reader to Kruskal's review of the Painlevé approach and its extensions: Kruskal, M. D.:Flexibility in Applying the Painlevé Test, in: D. Levi and P. Winternitz (ed.),Painlevé Transcendents, NATO ASI series B278, Plenum (1992) p. 175. Still, as far as the present review is concerned, this presentation of the Painlevé property will suffice.
5. Before proceeding further, let us make certain points clear. The term integrability will be used loosely, in this paper, to indicate integrability through IST techniques. It will, however, cover the cases where the nonlinear equation can be linearized into a purely differential one through a Cole-Hopf transformation. What it doesnot certainly cover are the cases where the system is reduced to quadratures. In this case there does not seem to exist any relation between the Painlevé property and integrability. This question was examined in detail in our course at les Houches, '89: Kruskal, M. D., Grammaticos, B., and Ramani, A.:Singularity Analysis and its Relation to Complete, Partial and Nonintegrability, NATO ASI C310 (1990) p. 321. While the whole Painlevé business can be formulated as a conjecture, we do not claim that a largeapplicability theorem can be obtained. There even exist counterarguments to that: Ramani, A., Grammaticos, B., and Yoshida, H.:Acta Appl. Math. 8 (1987), 75. However, our experience over more than a decade working on integrable systems has shown that once a system passes the Painlevé test it is most probably integrable. Counterexamples do exist but in every case one can point to some pathological feature of the system as an explanation. In this paper we shall not worry about these (marginal) cases but will focus on the use of the Painlevé criterion as an integrability predictor.