Some examples for the Poincaré and Painlevé problems

In 1891, Poincaré started a series of three papers in which he tried to answer the following question (cf. 21-23]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral). More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”. In this paper we give examples of one-parameter families which show that both problems have a negative answer. With some of the families we can also answer a question posed by M. Brunella in 5].