  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.
