Characterization of $n$-dimensional normal affine $SL_n$ -varieties

We show that any normal irreducible affine $n$-dimensional $SL_n$-variety $X$ is determined by its automorphism group in the category of normal irreducible affine varieties: if $Y$ is an irreducible affine normal algebraic variety such that $Aut(X) \cong Aut(Y)$ as ind-groups, then $Y \cong X$ as varieties. If we drop the condition of normality on $Y$ , then $X$ is not uniquely determined and we classify all such varieties. In case $n \ge 3$, all the above results hold true if we replace $Aut(X)$ by $U(X)$, where $U(X)$ is the subgroup of $Aut(X)$ generated by all one-dimensional unipotent subgroups. In dimension $2$ we have some very interesting exceptions.