The following is a variation on this question (or this question).
Let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \mapsto (x,-y)$, namely, $\beta$ is an automorphism of order two. Denote the set of symmetric elements w.r.t $\beta$ by $S_{\beta}$ and the set of skew-symmetric elements w.r.t $\beta$ by $K_{\beta}$.
Let $s_1,s_2 \in S_{\beta}$ and $k_1,k_2 \in K_{\beta}$. Write $p=s_1+k_1$ and $q=s_2+k_2$. Denote the ideal generated by $p$ and $q$ by $I$, $I= \langle p,q \rangle$.
Assume that $s_1,s_2,k_1,k_2 \in I$.
Should $\{s_1,s_2\}$ or $\{k_1,k_2\}$ be algebraically dependent? (see the comments in the linked question).
Clearly, $I$ is an invariant $k$-subspace, but I do not see how this helps. Perhaps this paper is relevant.
Edit: We can define $G:=\{1,\beta\}$ ($G$ is a group) and then $I$ is $G$-invariant. See this question.
Edid 2: Observe that, for example, we do not know if $J:=\langle p^2,q \rangle \subset I$ is invariant under $\beta$.
Any comments and hints are welcome! Now I have asked this question at MO.
