A quasi-smooth well-formed complete intersection log del Pezzo surface of index one admits an anticanonical polar cylinder if and only if it is the complete intersection of two quadrics in P4.
Araujo, K¨ahler-Einstein metrics for some quasi-smooth log del Pezzo surfaces, Trans
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Rigid affine cones over singular del Pezzo surfaces
A quasi-smooth well-formed complete intersection log del Pezzo surface of index one admits an anticanonical polar cylinder if and only if it is the complete intersection of two quadrics in P4.