  L. Makar-Limanov computed the automorphisms groups of surfaces in
$\mathbb{C}^{3}$ defined by the equations $x^{n}z-P(y)=0$, where $n\geq1$ and
$P(y)$ is a nonzero polynomial. Similar results have been obtained by A.
Crachiola for surfaces defined by the equations $x^{n}z-y^{2}-h(x)y=0$, where
$n\geq2$ and $h(0)\neq0$, defined over an arbitrary base field. Here we
consider the more general surfaces defined by the equations $x^{n}z-Q(x,y)=0$,
where $n\geq2$ and $Q(x,y)$ is a polynomial with coefficients in an arbitrary
base field $k$. Among these surfaces, we characterize the ones which are
Danielewski surfaces and we compute their automorphism groups. We study closed
embeddings of these surfaces in affine 3-space. We show that in general their
automorphisms do not extend to the ambient space. Finally, we give explicit
examples of $\mathbb{C}^{*}$-actions on a surface in $\mathbb{C}^{3}$ which can
be extended holomorphically but not algebraically to a $\mathbb{C}^{*}$-action
on $\mathbb{C}^{3}$.
