  Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group.
We study the natural action of $G$ on the space of isomorphism classes of right
ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right
$A_1$-modules). We show that this space breaks up into a countable number of
orbits each of which is a finite dimensional algebraic variety. Our results are
strikingly similar to those for the commutative algebra of polynomials in two
variables; however, we do not know of any general principle that would allow us
to predict this in advance. As a key step in the proof, we obtain a new
description of the bispectral involution of \cite{W1}. We also make some
comments on the group $G$ from the viewpoint of Shafaravich's theory of
infinite dimensional algebraic groups.
