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An $\\alpha$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $\\alpha$: $f \\alpha= \\alpha f$. It was shown that if $f$ is an $\\alpha$-endomorphism of $K[x,y]$ having a non-zero scalar Jacobian, then $f$ is invertible. 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