  We prove some new degeneracy results for integral points and entire curves on
surfaces; in particular, we provide the first example, to our knowledge, of a
simply connected smooth variety whose sets of integral points are never
Zariski-dense (and no entire curve has Zariski-dense image). Some of our
results are connected with divisibility problems, i.e. the problem of
describing the integral points in the plane where the values of some given
polynomials in two variables divide the values of other given polynomials.
