On a class of Danielewski surfaces in affine 3-space

Revised version with simplified proofs. A classification of special Danielewski surfaces admitting multiplicative group actions has been added.L. Makar-Limanov computed the automorphisms groups of surfaces in $\mathbb{C}^{3}$ defined by the equations $x^{n}z-P\left(y\right)=0$, where $n\geq1$ and $P\left(y\right)$ is a nonzero polynomial. Similar results have been obtained by A. Crachiola for surfaces defined by the equations $x^{n}z-y^{2}-h\left(x\right)y=0$, where $n\geq2$ and $h\left(0\right)\neq0$, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations $x^{n}z-Q\left(x,y\right)=0$, where $n\geq2$ and $Q\left(x,y\right)$ 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 $\mathbb{C}^{3}$