  We develop technics of birational geometry to study automorphisms of affine
surfaces admitting many distinct rational fibrations, with a particular focus
on the interactions between automorphisms and these fibrations. In particular,
we associate to each surface $S$ of this type a graph encoding equivalence
classes of rational fibrations from which it is possible to decide for instance
if the automorphism group of $S$ is generated by automorphisms preserving these
fibrations.
