On degree-2 Du Val del Pezzo surfaces admitting an (-K_S)-polar cylinder, an H-polar cylinder exists for every ample Q-divisor H.
Belousov, Cylinders in del Pezzo surfaces with du Val singularities , Bull
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Polarized cylinders in Du Val del Pezzo surfaces of degree two
On degree-2 Du Val del Pezzo surfaces admitting an (-K_S)-polar cylinder, an H-polar cylinder exists for every ample Q-divisor H.