Cylindricity of weighted singular del Pezzo surfaces over fields of characteristic zero

In this paper, we study the cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces obtained by blowing up weighted projective planes $\mathbb{P}(1,1,m)$ over an arbitrary field $\Bbbk$ of characteristic zero. As an application, we obtain vertical cylinders on higher-dimensional fibrations whose generic fibers are such $\Bbbk$-forms.