On simultaneous uniformization and local nonuniformizability

We prove existence of a one-dimensional holomorphic foliation (with isolated irremovable singularities) tangent to a rational vector field on appropriate affine algebraic surface of dimension 2 such that the family of leaves intersecting arbitrary given cross-section does not admit a uniformization holomorphic in the parameter by a family of simply connected domains in $\text{C}$. We show that such a foliation can be chosen transversally affine, having a Liouvillian first integral, with dense and hyperbolic leaves and an attracting cycle. This extends the author's result 4] giving a negative answer to Ilyashenko's simultaneous uniformization conjecture and answers negatively to the local version of this conjecture recently proposed by Shcherbakov. To cite this article: A. Glutsyuk, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 489–494.