  We classify up to isomorphism all finite-dimensional Lie algebras that can be
realised as Lie subalgebras of the complex Weyl algebra $A_1$. The list we
obtain turns out to be discrete and for example, the only non-solvable Lie
algebras with this property are:
  $sl(2)$, $sl(2)\times\mathbb C$ and $sl(2)\ltimes{\cal H}_3$.
  We then give several different characterisations, normal forms and isotropy
groups for the action of $Aut (A_1)\times Aut (sl(2))$ on a particular class of
realisations of $sl(2)$ in $A_1$.
