Embedding theorem for flexible varieties

Let $Z$ be an affine algebraic variety and $X$ be a smooth flexible variety. We develop some criteria under which $Z$ admits a closed embedding into $X$. In particular, we show that if $X$ is isomorphic (as an algebraic variety) to a special linear group and $\dim X\ge \max (2\dim Z+1,\dim TZ)$, then $Z$ admits a closed embedding into $X$.