  The two main results of this paper concern the regularity of the invariant
foliation of a C0-integrable symplectic twist diffeomorphisms of the
2-dimensional annulus, namely that $\bullet$ the generating function of such a
foliation is C1 ; $\bullet$ the foliation is H{\"o}lder with exponent 1/2. We
also characterize foliations by graphs that are straightenable via a symplectic
homeomorphism and prove that every symplectic homeomorphism that leaves
invariant all the leaves of a straightenable foliation has Arnol'd-Liouville
coordinates, in which the Dynamics restricted to the leaves is conjugated to a
rotation. We deduce that every Lipschitz integrable symplectic twist
diffeomorphisms of the 2-dimensional annulus has Arnol'd-Liouville coordinates
and then provide examples of 'strange' Lipschitz foliations in smooth curves
that cannot be straightened by a symplectic homeomorphism and cannot be
invariant by a symplectic twist diffeomorphism.This article is a part of
another preprint of the authors, entitled On the transversal dependence of weak
K.A.M. solutions for symplectic twist maps, after rewriting ant adding of the
H{\"o}lder part.
