  Let $\operatorname{Witt}$ be the Lie algebra generated by the set
$\{L_i\,\vert\, i \in {\mathbb Z}\}$ and $\operatorname{Vir}$ its universal
central extension. Let $\operatorname{Diff}(V)$ be the Lie algebra of
differential operators on $V=\mathbb{C}[[z]]$, $\mathbb{C}((z))$ or
$V=\mathbb{C}(z)$. We explicitly describe all Lie algebra homomorphisms from
$\mathfrak{sl}(2)$, $\operatorname{Witt}$ and $\operatorname{Vir}$ to
$\operatorname{Diff}(V)$ such that $L_0$ acts on $V$ as a first order
differential operator.
