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Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\\rm Spec}(\\mathfrak{R})$ admits an algebraic embedding in $\\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\\mathbb{A}_k^4$. 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