Lüroth's theorem for fields of rational functions in infinitely many permuted variables

Lüroth's theorem describes the dominant maps from rational curves over a field. We study those dominant rational maps from cartesian powers $X^{\Psi}$ of geometrically irreducible varieties $X$ over a field $k$ for infinite sets $\Psi$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h\colon X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\setminus Y^{\Psi}$, where $X\xrightarrow{f}Y$ is a dominant $k$-map and $H$ is a group of birational automorphisms of $Y|k$, acting diagonally on $Y^{\Psi}$.In characteristic 0 we show that this construction, when properly modified, produces all dominant equivariant maps from $X^{\Psi}$ in the case $\dim X=1$. For arbitrary $X$ some partial results are obtained.Also, a similar problem of the description of equivariant integral schemes over $X^{\Psi}$ of finite type is touched very briefly.