  L\"uroth's theorem describes the dominant maps from rational curves over a
field.
  In this note we study the dominant maps from cartesian powers $X^{\Psi}$ of
absolutely 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
$h:X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\backslash Y^{\Psi}$, where
$X\xrightarrow{f}Y$ is a dominant $k$-map and $H$ is an automorphism group $H$
of $Y|k$, acting diagonally on $Y^{\Psi}$.
  In characteristic 0, we show that this construction, when properly modified,
gives all dominant equivariant maps from $X^{\Psi}$, if $\dim X=1$. For
arbitrary $X$, the results are only partial.
  In a subsequent paper, the `quasicoherent' equivariant sheaves on the targets
of such $h$'s will be studied. Some preliminary results have already appeared
in arXiv:math/2205.15144.
  A somewhat similar problem is to check, whether the irreducible invariant
subvarieties of $X^{\Psi}$ arise as pullbacks under $f^{\Psi}$ (for appropriate
$f$'s) of subvarieties of $Y$ diagonally embedded into $Y^{\Psi}$. This would
be a complement to the famous theorem of D.E.Cohen on the noetherian property
of the symmetric ideals. We show that this is the case if $\dim X=1$.
