Let $k$ be a field of characteristic zero, $\beta$ an involution on $k[x,y]$ (a $k$-algebra automorphism of $k[x,y]$ of order two), and $U \subseteq k[x,y]$ an ideal of $k[x,y]$ (a $k[x,y]$-submodule of $k[x,y]$).
Does $\beta(U) \subseteq U$ imply something interesting about $U$?
Examples for 'interesting': (i) $\dim(U) < \infty$ ($\dim$ may refer to any relevant dimension). (ii) There exists a generating set for $U$ as a $k[x,y]$-module consisting of 'pure' elements (pure = symmetric or skew-symmetric).
See also this.
Edit: More specifically, I am interested in a two generated ideal $U=\langle p,q \rangle$. Hopefully, considering such $U$ as a $k$-subspace or a $k[x,y]$-module may yield something interesting.
Any comments and hint are welcome!