Let $X$ be the complex Fermat quartic surface defined by the polynomial $x^4+y^4+z^4+w^4$.

By results of Sertöz, we know that the surface $X$ admits at least one Enriques involution, i.e. an involution without fixed points. Is an explicit formula for one such involution already known?

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    $\begingroup$ A remark: any biregular involution of $\mathbb{P}^3$ has two lines of fixed points, hence (generically) eight fixed points on $X$. So one cannot hope to get an Enriques involution by the restriction of a biregular involution of the ambient space. I do not know if there is a construction using a birational involution of $\mathbb{P}^3$. $\endgroup$ – Francesco Polizzi Feb 6 at 10:46

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