Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry $g\in O(L)$ such that $g^i$ is fixed point free on $L$ for any $1\le i\le \mathrm{\mid }g\mathrm{\mid }-1$. We show that when $L_2=\mathrm{\varnothing }$ and $g^i$ is fixed point free on $L$ for any $1\le i\le \mathrm{\mid }g\mathrm{\mid }-1$, $V_L^{\widehat{g}}$ has extra automorphisms implies either (1) the order of $g$ is a prime or (2) $L$ is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.