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