ML-invariant and automorphism groups of certain word varieties in SL0.55542pt(2,C)

For a fixed element g∈ SL0.55542pt(2,C) g ∈ SL ( 2 , C ) and a word w=[xⁿ\!,yᵐ] w = [ x n , y m ] we consider the automorphism group Aut0.55542pt(Sg) Aut ( S g ) of the affine threefold Sg=\(x,y)∈ SL0.55542pt(2,C)² \, |\, w(x,y)=g\ S g = { ( x , y ) ∈ SL ( 2 , C ) 2 | w ( x , y ) = g } . We prove that the Makar-Limanov invariant ML(Sg)=O(Sg) ML ( S g ) = O ( S g ) and Aut0.55542pt(Sg) Aut ( S g ) is Jordan.