 the following proposition.  If $X\in {\mathfrak{X}^*_m(M)}$ has no singularities then there exist a neighborhood ${{\mathcal U}}$ of $X$ in ${\mathfrak{X}^0_m(M)}$ and $i\in {\mathbb{N}}$, such that the index of any periodic orbit of any vector field $Y\in {{\mathcal U}}$ is $i$. 