Involutions on $\mathbb{C}(x,y)$ How to find all involutions on $\mathbb{C}(x,y)$,
or at least all involutions $\delta$ on $\mathbb{C}(x,y)$ such that
$\delta(x)=x$?
Remarks:
(1) An involution on $\mathbb{C}[x,y]$ is either conjugate to $\beta: (x,y) \mapsto (x,-y)$ or to $\epsilon: (x,y) \mapsto (-x,-y)$
(since the group of automorphisms of $k[x,y]$ is free amalgamated, and by a result of Serre about trees).
(2) In $\mathbb{C}[x,y]$ if we wish to find all involutions $\delta$ with $\delta(x)=x$,
just write $\delta(y)=q$. There are two cases: 
(a) The Jacobian of $\delta$ is $1$, so $q_y=1$, and then
$q=y+H(x)$, for some $H(t) \in \mathbb{C}[t]$. 
Since $\delta$ is of order $2$, by $\delta^2=1$ (=order $\leq 2$),
we get $h(x)=0$, so $\delta=1$, which is not an involution.
(b) The Jacobian of $\delta$ is $-1$, so $q_y=-1$, and then
$q=-y+H(x)$, for some $H(t) \in \mathbb{C}[t]$.
Since $\delta$ is of order $2$, by $\delta^2=1$ (=order $\leq 2$),
we get that there is no restriction on $h(x)$, so
$\delta: (x,y) \mapsto (x,-y+H(x))$. (Another way to obtain this is by using  Theorem 1).
(3) Of course, in $\mathbb{C}(x,y)$ there exist more involutions, for example,
$(x,y) \mapsto (x,\frac{1}{y})$. Is it true that the $\delta$'s I am looking for are those of (2)(b), $(x,y) \mapsto (x,\frac{1}{y})$ and 
$(x,y) \mapsto (x,-\frac{1}{y})$?
(4) I have asked this question, 
in which commenters suggested the following references: Serge Cantat's notes and 
Julie Deserti's list. (Perhaps I can find an answer to my recent question in those references).
Thank you very much.
 A: As you explain, the classification of birational involutions of $\mathbb{P}^2$ given by Bayle and Beauville in the article you cite gives the answer.
The Jonquières involutions preserve a rational fibration. They are in fact conjugate to $(x,y)\mapsto (x,p(x)/y)$ for some polynomial $p\in \mathbb{C}$ without any multiple root and of even degree. The fixed locus is given by the curve $y^2=p(x)$, which is rational if $\deg(p)=2$, elliptic is $\deg(p)=4$ and hyperelliptic of genus $g$ when $\deg(p)=2g+2\ge 6$. The birational type of the curve determines the conjugacy class in the Cremona group $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$. In particular, for $\deg(p)=2$ you have only one conjugacy class, in fact also conjugate to $(x,y)\mapsto (x,-y)$ in $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$ (but not by an element that fixes $x$).
The other involutions (Geiser and Bertini) do not preserve any rational fibration. They preserve elliptic fibrations, but this does not help so much to write explicitely the map. This can be done in practice but the formulas are quite horrible. In particular, the degree of a Bertini involution is $17$ and the degree of a Geiser Bertini involution is $8$. So $x$ and $y$ are sent onto rational functions, quotients of polynomials of degree $17$ or $8$. Choosing the coordinates such that the point $[1:0:0]$ or $[0:1:0]$ (pencils of $x/z$ and $y/z$ corresponding to $x=cst$ and $y=cst$) you obtain polynomials of degree slightly less ($11$ and $5$) but it not so good for expliciting the maps. The geometric description given by Del Pezzo surfaces of degree $1$ and $2$ that you can find in the article of Bayle and Beauville is definitely better, I think.
