ML-invariant and automorphism groups of certain word varieties in SL(2,C)^2

For a fixed element $g\in SL(2,C)$ and a word $w=[x^n,y^m]$ we consider the automorphism group $Aut(S_g)$ of the affine threefold $S_g=\{(x,y)\in SL(2,C{)}^2\mathrm{\mid }w(x,y)=g\}\mathrm{.}$ We prove that Makar-Limanov invariant $ML(S_g)=\mathcal{O}(S_g)$ and $Aut(S_g)$ is Jordan.