Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $F_{-1} : k[x,y] \to R_{-1}$ be an injective $k$-algebra homomorphism, and denote $A_{-1}:=F_{-1}(x)$, $B_{-1}:=F_{-1}(y)$.
$A_{-1},B_{-1}$ should be algebraically independent over $k$, by the injectivity of $F_{-1}$, if I am not wrong.
(1) Is it true that the Jacobian of $A_{-1},B_{-1}$ belongs to $k^{\times}=k-\{0\}$?
For example: $A_{-1}=x, B=y+x^{-1}$ 'seems' an injective homomorphism.
(2) Is there something 'interesting' that can be said about such an homomorphism? (except somethimg about the Jacobian),
An example for something 'interesting': The $(1,-1)$-degree of each of $A_{-1},B_{-1}$ must be $\geq -1$.
Remarks:
(i) More generally, we can ask the same question, replacing $R_{-1}$ by $R_{n,-n}:=k[x,x^{\frac{1}{n}},x^{-\frac{1}{n}},y]$; see this relevant question.
(ii) Similar questions (Jacobian replaced by the commutator) can be asked for the first Weyl algebra $A_1(k)$ instead of for $k[x,y]$, with an appropriate relation $[y,x^{\frac{1}{n}}]=\ldots$. Notice that now injectivity is automatic, since $A_1(k)$ is simple.
Any hints and comments are welcome!