Killing vector fields of constant length on compact hypersurfaces

We show that if a compact hypersurface $M \subset \mathbb{R}^{n+1}$, $n \geq3$, admits a non zero Killing vector field $X$ of constant length then $n$ is even and $M$ is diffeomorphic to the unit hypersphere of $\mathbb{R}^{n+1}$. Actually, we show that $M$ is a complex ellipsoid in $\mathbb{C}^{N} = \mathbb{R}^{n+1}$. As an application we give a simpler proof of a recent theorem due to S. Deshmukh \cite{De12}.