Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). I am looking for examples of rank one, degree zero reflexive sheaf $F$ such that there exists an integer $r>0$ for which there is no rank one reflexive sheaf $G$ for which $G^{\otimes r}=F$. Is there some general strategy to produce such examples? Moreover, what happens if $F$ is an invertible sheaf?

Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (degree under the fixed polarisation) reflexive sheaf $F$ such that there exists an integer $r>0$ for which there is no rank one reflexive sheaf $G$ for which $G^{\otimes r}=F$. Is there some general strategy to produce such examples? Moreover, what happens if $F$ is an invertible sheaf?