Non-symplectic automorphisms of K3 surfaces with one-dimensional moduli space

The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $\\sigma$ order $n\\geq 2$ is one dimensional exactly when $\\varphi(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,\\sigma)$ irreducible components maximal dimension such spaces. particular, show that there unique one-dimensional component $n=20,22, 24$, three $n=15$ two in remaining cases.