Bracket Width of the Lie Algebra of Vector Fields on a Smooth Affine Curve

We prove that the bracket width of the simple Lie algebra of vector fields Vec(C) of a smooth irreducible affine curve C with a trivial tangent sheaf is at most three.In addition, if C is a plane curve, the bracket width of Vec(C) is at most two and if moreover C has a unique place at infinity, the bracket width of Vec(C) is exactly two.We also show that in case C is rational, the width of Vec(C) equals one.