Let $f=f(x,y), g=g(x,y) \in k[x,y]$ be two non-constant polynomials in two variables over a field of characteristic zero $k$.
Assume that $k[f,g,x]=k[x,y]$ and that $Jac(f,g):=f_xg_y-f_yg_x \in k[x,y]-k$ (namely, $f$ and $g$ are algebraically independent over $k$, but are not a Jacobian pair, where a Jacobian pair is a pair of polynomials having a Jacobian in $k^*$). For example: $f=xy$ and $g=y$ (here, $Jac(f,g):=f_xg_y-f_yg_x=y1-x0=y$).
Can one find a description for all such pairs of polynomials?
A partial answer is: $(f,g)=(f,y)$ with $f_x \in k[x,y]-k$; indeed, $k[f,g,x]=k[f,y,x]=k[x,y]$ and $Jac(f,g)=f_x1-f_y0=f_x$.
Remark: If we replace the assumption that $Jac(f,g) \in k[x,y]-k$ by the assumption that $Jac(f,g) \in k^*$, then the answer is as follows: By Formanek's theorem, $k[f,g,x]=k[x,y]$ and $Jac(f,g) \in k^*$ imply that $k[f,g]=k[x,y]$. Then, $(x,y) \mapsto (f,g)$ defines an automorphism of $k[x,y]$ and it is well-known that an automorphism of $k[x,y]$ is a products of affine and triangular automorphisms. (Another description: The group of automorphisms of $k[x,y]$ is a certain free amalgamated group).
Thank you very much!