Let $X$ be a (smooth) del Pezzo surface over $\mathbb{C}$. Let $\Delta_0$ be a (smooth irreducible) generic curve in the linear system $|-2K_X|$. Let $\rho : S \rightarrow X$ be the double cover of $X$ branched over $\Delta_0$ and let $i$ be the associated involution on $S$. Let $\Delta$ be the branching curve of $\rho$, then $\Delta \in |\rho^*(-K_X)|$ is a fixed point for the action of $i^*$ on the linear system $|\rho^*(-K_X)|$.

I would like to know if it is always an **isolated** fixed point for the action of $i^*$ on $|\rho^*(-K_X)|$? I am reading a paper where it is stated (without proof) to be true if $X$ is a del Pezzo surface of degree $2$. I don't know how to prove it in the case of the del Pezzo of degree $2$ and I'd like to know if it may be true for other Del Pezzo surfaces.