  We study the solvability in $L^p$ of the $\bar\partial$-equation in a
neighborhood of a canonical singularity on a complex surface, a so-called du
Val singularity. We get a quite complete picture in case $p=2$ for two natural
closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For
$\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is
solvability if and only if a certain boundary condition $(*)$ is fulfilled at
the singularity. Our main tool is certain integral operators for solving
$\bar\partial$ introduced by the first and fourth author, and we study mapping
properties of these operators at the singularity.
