  For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of
$Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle
equivalent to the singularity category of Buchweitz and Orlov. Its thick
subcategory given as the stable category of $CM_0^ZR$ is central in
representation theory since it enjoys Auslander-Reiten-Serre duality and has
almost split triangles. In the case $dim R=1$, we prove that the stable
category of $CM_0^ZR$ always admits a silting object, and that it admits a
tilting object if and only if either $R$ is regular or the $a$-invariant of $R$
is non-negative.
