Maximal Haagerup subalgebras in L(Z2?SL2(Z))

We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof handle $\textbf{k}=\mathbb{Z}$. The key step complete description of all intermediate subalgebras between and $L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where $SL_2(\textbf{k})\curvearrowright Y$ denotes quotient algebraic action \widehat{\textbf{k}^2}$ by modding out relation $\phi\sim \phi'$, $\phi$, $\phi'\in $\phi'(x, y):=\phi(-x, -y)$ $(x, y)\in \textbf{k}^2$. As by-product, $L(PSL_2(\mathbb{Q}))$ PSL_2(\mathbb{Q})$; particular, $PSL_2(\mathbb{Q})\curvearrowright prime action, i.e. it admits no non-trivial actions.