Involutions and the Jacobian conjecture

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $\alpha$ be the exchange involution on $K[x,y]$: $\alpha (x)=y$ and $\alpha (y)=x$. 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. Based on this, we bring more results that imply that a given endomorphism $f$ having a non-zero scalar Jacobian and additional conditions involving involutions, is invertible.