  Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over
a characteristic zero field $K$ and let $\alpha$ be the exchange involution on
$A_1(K)$ given by $\alpha(x)= y$ and $\alpha(y)= x$. The Dixmier conjecture of
Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$
an automorphism? The aim of this paper is to prove that each
$\alpha$-endomorphism of $A_1(K)$ is an automorphism. Here an
$\alpha$-endomorphism of $A_1(K)$ is an endomorphism which preserves the
involution $\alpha$. We also prove an analogue result for the Jacobian
conjecture in dimension 2, called $\alpha-JC_2$.
