I am looking for ideals $I\subset \mathbb{F}\_2[x,y]$ with the following properties:


1. $I$ is generated by two homogeneous elements;
2. $I$ is invariant under the $SL\_2(\mathbb{F}\_2)$-action on $\mathbb{F}\_2[x,y]$ (given by extending the action on the two dimensional sub vector space spanned by $x,y$).;
3. The quotient $\mathbb{F}\_2[x,y]/I$ is finite.


So far the only examples I know are $(x^n,y^n)$ and $(x^3,x^2+xy+y^2)$. The quotient $\mathbb{F}\_2[x,y]/(x^3,x^2+xy+y^2)$ is the cohomology ring $H^\*(S^3/Q\_8;\mathbb{F}\_2)$ for the standard action of the quaternion group on the three sphere. 


***Are these the only examples. Is it possible to classify all such ideals ?***


Edit: Of course $(x^n,y^n)$ is only invariant when $n$ is a power of two (since $x\mapsto x, y\mapsto x+y$ is also an automorphism). So we have even fewer examples. 



Ideals invariant under ring automorphisms
There are tons of examples. Put $u = x^2+xy+y^2$ and $v = xy(x+y)$. Then $u$ and $v$ are $SL\_2(\mathbb{F}\_2)$ invariant. If $f(s,t)$ and $g(s,t)$ are homogenous polynomials with respect to the grading $\deg s = 2$, $\deg t=3$, then $\langle f(u,v), g(u,v) \rangle$ is a $SL\_2(\mathbb{F}\_2)$-invariant generated in the manner you describe. I claim that, furthermore, if $\mathbb{F}\_2[s,t]/\langle f, g \rangle$ is finite then so is $\mathbb{F}\_2[x,y] / \langle f(u,v), g(u,v) \rangle$.


For this claim, we just need to show that $\mathbb{F}\_2[x,y]$ is module finite over $\mathbb{F}\_2[u,v]$. It is enough to show that the generators $x$ and $y$ are integral over $\mathbb{F}\_2[u,v]$. Indeed, they are both roots of the monic polynomial 
$$z^3 + u z + v.$$
(I found $u$ and $v$ by multiplying out the polynomial $(z+x)(z+y)(z+x+y)$.)


