1. See, e.g., M. D. Kruskal, A. Ramani, and B. Grammaticos, Singularity analysis and its relation to complete, partial and non-integrability, in Partially integrable nonlinear evolution equations and their physical applications, based on lectures given at the Les Houches Spring School (March 21–30, 1989), ed. R. Conte (1990); also P. A. Clarkson and M. D. Kruskal, to appear in Studies in Applied Mathematics; also several other articles in the present volume; and of course references in all of these.
2. See §3.5 of Kruskal, Ramani, and Grammaticos, loc. cit.
3. See p. 2 of P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont Vintégrale générale est uniforme, Acta Mathematica 25, pp. 1-85 (1902).
4. See p. 204 of P. Painlevé, Mémoire sur les équations différentielles dont Vintégrale générale est uniforme, Bull. Soc. Math. France 28, pp. 201–261 (1900).
5. S. Kovalevskaya seems actually to have been first to use the test as we do today, in her Bordin-Prize-winning paper Mémoire sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Mathematica 12, pp. 177-232 (1889).