  We study the group of birational transformations of the plane that fix (each
point of) a curve of geometric genus 1.
  A precise description of the finite elements is given; it is shown in
particular that the order is at most 6, and that if the group contains a
non-trivial torsion, the fixed curve is the image of a smooth cubic by a
birational transformation of the plane.
  We show that for a smooth cubic, the group is generated by its elements of
degree 3, and prove that it contains a free product of Z/2Z, indexed by the
points of the curve.
