  We consider foliations of compact complex manifolds by analytic curves. We
suppose that the line bundle tangent to the foliation is negative. We show that
in a generic case there exists a finitely smooth homeomophism, holomorphic on
the fibers and mapping fiberwise the manifold of universal coverings over the
leaves passing through some transversal $B$ onto some domain in
$B\times\mathbb{C}$ with continuous boundary. depending on the leaves. The
problem can be reduced to a study of the Beltrami equation with parameters on
the unit disk in the case, when derivatives of the corresponding coefficient
Beltrami grow no faster than some negative power of the distance to the
boundary of the disk.
