  Let $k$ be a finitely generated field, let $X$ be an algebraic variety and
$G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$
and every element of a Zariski-dense semigroup $\Gamma \subset G(k)$ has a
rational fixed point in $X(k)$. We then deduce, under some mild technical
assumptions, the existence of a rational map $G\to X$, defined over $k$,
sending each element $g\in G$ to a fixed point for $g$. The proof makes use of
a recent result of Ferretti and Zannier on diophantine equations involving
linear recurrences. As a by-product of the proof, we obtain a version of the
classical Hilbert Irreducibility Theorem valid for linear algebraic groups.
