  A birationally liftable Galois section s of a hyperbolic curve X/k over a
number field k yields an adelic point x(s) in the smooth completion of X. We
show that x(s) is X-integral outside a set of places of Dirichlet density 0, or
s is cuspidal. The proof relies on $GL_2(F_\ell)$-quotients of $\pi_1(U)$ for
some open U of X.
  If k is totally real or imaginary quadratic, we prove that all birationally
adelic, non-cuspidal Galois sections come from rational points as predicted by
the section conjecture of anabelian geometry. As an aside we also obtain a
strong approximation result for rational points on hyperbolic curves over Q or
imaginary quadratic fields.
