  The well-known Dixmier conjecture asks if every algebra endomorphism of the
first Weyl algebra over a characteristic zero field is an automorphism. We
bring a hopefully easier to solve conjecture, called the $\gamma,\delta$
conjecture, and show that it is equivalent to the Dixmier conjecture. Up to
checking that in the group generated by automorphisms and anti-automorphisms of
$A_1$ all the involutions belong to one conjugacy class, we show that every
involutive endomorphism from $(A_1,\gamma)$ to $(A_1,\delta)$ is an
automorphism ($\gamma$ and $\delta$ are two involutions on $A_1$), and given an
endomorphism $f$ of $A_1$ (not necessarily an involutive endomorphism), if one
of $f(X)$,$f(Y)$ is symmetric or skew-symmetric (with respect to any involution
on $A_1$), then $f$ is an automorphism.
