Killing vector fields of constant length on compact hypersurfaces

We show that if a compact hypersurface $M$ , $n > 2$ of the Euclidean space, admits a non zero Killing vector field $X$ of constant length then $n$ is odd and $M$ is diffeomorphic to the unit hypersphere of $R^{n+1}$. Actually, we show that $M$ is a complex ellipsoid in $C^N = R^{n+1}$. As an application we give a simpler proof of a recent theorem due to S. Deshmukh.