# Form of $u,v \in \mathbb{C}[x,y]$ satisfying $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$

Let $$\beta$$ be an involution on $$\mathbb{C}[x,y]$$, namely, $$\beta$$ is a $$\mathbb{C}$$-algebra automorphism of $$\mathbb{C}[x,y]$$ of order two. Denote the set of symmetric elements with respect to $$\beta$$ by $$S_{\beta}$$ and the set of skew-symmetric elements w.r.t. $$\beta$$ by $$K_{\beta}$$.

Let $$s_1,s_2 \in S_{\beta}$$ and $$k_1,k_2 \in K_{\beta}$$. Assume that every two of $$\{s_1,s_2,k_1,k_2\}$$ are algebraically independent over $$\mathbb{C}$$. Then $$\mathbb{C}(s_1,s_2,k_1,k_2)$$ has transcendence degree two over $$\mathbb{C}$$.

By a known result from algebraic geometry (which appears here), there exist $$u,v \in \mathbb{C}(x,y)$$ such that $$\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$$.

Assume, for simplicity, that $$u,v \in \mathbb{C}[x,y]$$ (perhaps it is possible to find such $$u,v$$, see this question).

Call Property P the following property: There exist an automorphism $$g$$ of $$\mathbb{C}[x,y]$$ such that $$g(u),g(v) \in S_{\beta} \cup K_{\beta}$$.

Could one find a (counter)example in which property P is not satisfied? Or a proof that property P is satisfied?

This question seems slightly relevant (perhaps we should assume that some field extension here is Galois?).

Thank you very much!