Let $\iota$ be an involution on $\mathbb{C}(x,y)$, namely, a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ of order two.
Examples of involutions: $\alpha: (x,y) \mapsto (y,x)$, $\beta: (x,y) \mapsto (x,-y)$, $\epsilon: (x,y) \mapsto (-x,-y)$. Observe that $\alpha$ and $\beta$ are conjugate, while $\epsilon$ is not conjugate to them. (The Jacobian of $\alpha$ and $\beta$ is $-1$, while the Jacobian of $\epsilon$ is $1$).
Assume that $u,v \in \mathbb{C}(x,y)$ satisfy $\iota(\mathbb{C}(u,v)) \subseteq \mathbb{C}(u,v)$.
Is it possible to characterize such $\mathbb{C}(u,v)$?
Same question with $u,v \in \mathbb{C}[x,y]$ instead of $u,v \in \mathbb{C}(x,y)$, but still with $\iota(\mathbb{C}(u,v)) \subseteq \mathbb{C}(u,v)$, not $\iota(\mathbb{C}[u,v]) \subseteq \mathbb{C}[u,v]$.
An example for a characterization of $\mathbb{C}(u,v)$ is as follows: There exists an automorphism $g$ of $\mathbb{C}[x,y]$ such that $g(u),g(v) \in S_{\iota} \cup K_{\iota}$, where $S_{\iota}$ is the set of symmetric elements of $\mathbb{C}(x,y)$ w.r.t. $\iota$ and $K_{\iota}$ is the set of skew-symmetric elements of $\mathbb{C}(x,y)$ w.r.t. $\iota$.
Examples:
(1) For $\alpha$ take $u=x^2-y^2$, $v=xy$. Clearly, $\alpha(\mathbb{C}(x^2-y^2,xy)) \subseteq \mathbb{C}(x^2-y^2,xy)$. We can take $g=1$; indeed, $u=x^2-y^2 \in K_{\alpha}$, $v=xy \in S_{\alpha}$.
(2) For $\beta$ take $u=x+y^2$, $v=x+y+y^2$. Clearly, $\mathbb{C}(x+y^2,x+y+y^2)=\mathbb{C}(x,y)$ which is $\beta$-invariant. $u \in S_{\beta} \subset S_{\beta} \cup K_{\beta}$, $v \notin S_{\beta} \cup K_{\beta}$.
Take the following automorphism $g: (x,y) \mapsto (x-(y-x)^2,y-x)$ and get $g(u),g(v) \in S_{\beta} \cup K_{\beta}$; indeed, $g(u)=g(x+y^2)=g(x)+g(y)^2=x-(y-x)^2+(y-x)^2=x \in S_{\beta}$ and $g(v)=g(x+y+y^2)=g(u+y)=g(u)+g(y)=x+(y-x)=y \in K_{\beta}$.
(3) More generally: If $h: (x,y) \mapsto (u,v)$ is an automorphism of $\mathbb{C}[x,y]$, then $\mathbb{C}[x,y]=\mathbb{C}[u,v]$, so $\mathbb{C}(x,y)=\mathbb{C}(u,v)$ is $\beta$-invariant, and taking $g:=h^{-1}$ we obtain $g(u)=h^{-1}(u)=h^{-1}(h(x))=x \in S_{\beta}$ and $g(v)=h^{-1}(v)=h^{-1}(h(y))=y \in K_{\beta}$.
I have also asked the above question in MSE.
This question is slightly relevant.
Any hints and comments are welcome!