I am trying to prove something for function fields in two generators and only one involution but I don't know if it is true, if true, I would like to generalize, here it is.

Let $K=k(\eta_1,\eta_2)$ be a function field, and let $\sigma$ be an involution such that $\sigma|_k=id$, how can I describe the generators of $K^{\langle \sigma \rangle}$?

If $\tau_1=\eta_1+\sigma\eta_1\neq 0$ and $\tau_2=\eta_2+\sigma\eta_2\neq 0$, When is true that $K^{\langle\sigma\rangle}=k(\tau_1,\tau_2)$ ?

What I can tell is that if $\lambda_1=\eta_1\cdot\sigma\eta_1$ and $\lambda_2=\eta_2\cdot\sigma\eta_2$ and of course $\eta_i^2-\tau_i\eta_i+\lambda_i=0$ for $i\in\lbrace 1,2\rbrace$ then $k(\tau_1,\tau_2,\lambda_1,\lambda_1)\subset k(\eta_1,\eta_2)$ is a field extension of degree $1$ or $2$ or $4$, so, Can I get a description of $K^{\langle\sigma\rangle}$?