Closed embeddings of C∗ in C2, part I

We consider closed curves C≃C∗ in the affine plane S≃C2 that admit a good or very good asymptote. By this we mean a curve L≃C in S that in suitable coordinates for S is linear and tangent to C at infinity, and meets C once or not at all at finite distance. We classify these curves up to automorphism of S. Relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of C at infinity and then proceed to give explicit equations.