# K3 surfaces with small Picard number and symmetry

I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution. Here, low is no mathematically precise concept. I want to do computations with Monad bundles and for that lower is better for me. For Picard rank bigger than 5 those computations become too difficult for me.

For example, the very general branched double cover of $$\mathbb{CP}^2$$ branched over a sextic has $$Pic \simeq \mathbb{Z}$$ and the map that swaps the two sheets of the cover is a holomorphic involution.

The post Picard groups of quartic K3 surfaces contains more examples of K3 surfaces with Picard ranks 1, 2, and 3, but I didn't find any holomorphic involutions of the K3 surfaces mentioned there.

there are many examples of complex elliptic K3 surfaces $$X$$ with Picard rank 3 and having (infinite) automorphism group containing involutions (in fact, $$\operatorname{Aut}(X)$$ contains a copy of $$\mathbb{Z}/2 \ast \mathbb{Z}/2$$).