  Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed.
Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$
where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a
$k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\
x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d
\in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let
$L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$.
We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of
a certain surface in $P^4_k$ which is given as the intersection of two
quadrics. We give criteria for the $k$-rationality of
$L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the
paper, we also give an alternative geometric proof of a part of the result
which is provided to the authors by J.-L. Colliot-Th\'el\`ene.
