Simple birational extensions of the polynomial algebra $\mathbb{C}^{[3]}$

The Abhyankar-Sathaye Problem asks whether any biregular embedding $\varphi:\mathbb{C}^k\hookrightarrow\mathbb{C}^n$ can be rectified, that is, whether there exists an automorphism $\alpha\in{\operatorname{Aut}}\,\mathbb{C}^n$ such that $\alpha\circ\varphi$ is a linear embedding. Here we study this problem for the embeddings $\varphi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4$ whose image $X=\varphi(\mathbb{C}^3)$ is given in $\mathbb{C}^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in\mathbb{C}[x,y]\backslash\{0\}$ and $g\in\mathbb{C}[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $\mathbb{C}^{[4]}=\mathbb{C}[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $\mathbb{C}^4$). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings $\mathbb{C}^2\hookrightarrow\mathbb{C}^3$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial $p$ as above, a criterion for when $X=p^{-1}(0)\simeq\mathbb{C}^3$.