The structure of algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$ (the normal quartic hypersurface case. II)

We obtain the affirmative answer for a special case of the linearization problem for algebraic embeddings of C 2 into C 3 .Indeed, we determine all the compactifications (X , Y ) of C 2 such that X are normal quartic hypersurfaces in P 3 without triple points and Y are hyperplane sections of X.Moreover, for each (X , Y ), we construct a tame automorphism of C 3 which transforms the hypersurface X n Y onto a coordinate hyperplane.