  At the end of 1960-ths Yu.S.Ilyashenko stated the problem: is it true that
for any one-dimensional holomorphic foliation with singularities on a Stein
manifold leaves intersecting a transversal disc can be uniformized so that the
uniformization function would depend holomorphically on the transversal
parameter? In the present paper we construct a two-dimensional counterexample.
This together with the previous result of Yu.S.Ilyashenko (Lemma 1) implies
existence of a counterexample given by a foliation on affine (projective)
algebraic surface by level curves of a rational function with singularities
deleted. This implies also that Bers' simultaneous uniformization theorem for
topologically trivial holomorphic foliations by compact Riemann surfaces does
not extend to the general fibrations by compact Riemann surfaces with
singularities.
