# Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$

Let $$r \in \mathbb{N}-\{0\}$$.

Commutative case: Let $$f : (x,y) \mapsto (p,q)$$ be a map from $$\mathbb{C}[x,y]$$ to $$\mathbb{C}[x^{1/r},x^{-1/r},y]$$ satisfying the following two conditions:

(i) $$\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$$.

(ii) $$l_{1,-1}(p)=\lambda x^{\frac{n}{r}+1}y$$ and $$l_{1,-1}(q)=\mu x^{-\frac{n}{r}}$$, where $$n \in \mathbb{Z}-\{0\}$$ and $$\lambda,\mu \in \mathbb{C}-\{0\}$$.

Then we can write $$p=\lambda x^{\frac{n}{r}+1}y + A$$ and $$q=\mu x^{-\frac{n}{r}}+B$$, where $$v_{1,-1}(A) < \frac{n}{r}$$ and $$v_{1,-1}(B) < -\frac{n}{r}$$.

Question 1: Is necessarily $$A \in k[x^{1/r},x^{-1/r}]$$ and $$B=0$$?

Non-commutative case:

Let $$A_1(\mathbb{C})$$ be the first Weyl algebra, namely, the associative non-commutative $$k$$-algebra generated by $$x$$ and $$y$$ subject to the relation $$yx-xy=1$$. Let $$A_1(\mathbb{C})^{-\frac{1}{r}}$$ be a ring extension of the first Weyl algebra, generated by $$x^{\frac{1}{r}}$$, $$x^{-\frac{1}{r}}$$ and $$y$$ subject to the relations $$yx-xy=1$$, $$x (x^{-\frac{1}{r}})^r= (x^{-\frac{1}{r}})^r x=1$$, and $$[y,x^{-\frac{1}{r}}]:=-\frac{1}{r} x^{-\frac{1}{r}-1}$$.

Let $$f : (x,y) \mapsto (p,q)$$ be a map from $$A_1(\mathbb{C})$$ to $$A_1(\mathbb{C})^{-\frac{1}{r}}$$ satisfying the following two conditions:

(i) $$[q,p]:=qp-pq \in \mathbb{C}-\{0\}$$.

(ii) $$l_{1,-1}(p)=\lambda x^{\frac{n}{r}+1}y$$ and $$l_{1,-1}(q)=\mu x^{-\frac{n}{r}}$$, where $$n \in \mathbb{Z}-\{0\}$$ and $$\lambda,\mu \in \mathbb{C}-\{0\}$$.

Then we can write $$p=\lambda x^{\frac{n}{r}+1}y + A$$ and $$q=\mu x^{-\frac{n}{r}}+B$$, where $$v_{1,-1}(A) < \frac{n}{r}$$ and $$v_{1,-1}(B) < -\frac{n}{r}$$.

Question 2: Is necessarily $$A \in k[x^{1/r},x^{-1/r}]$$ and $$B=0$$?

Remark: A special case of the above question can be found here (actually, the later edit of that question is the same as the above question).

Thank you veru much!