Cylinders in singular del Pezzo surfaces

For each del Pezzo surface with du Val singularities, we determine whether it admits a -polar cylinder or not. If it allows one, then we present an effective -divisor that is -linearly equivalent to and such that the open set is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial -actions on their affine cones defined by their anticanonical divisors.