Let $B$ be a Noetherian regular local ring with algebraically closed residue
field $k$, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime
order acting trivially on $k$. Let $A$ be the ring of $G$-invariants of $B$,
assume that $A$ is Noetherian. We study conditions under which $A$ is again
regular; in particular, we prove that $B$ is a monogenous $A$-algebra if and
only if $G$ is a generalized pseudo-reflection.