Improving the regularity of vector fields

Let $\alpha>0$, $\beta>\alpha$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{\alpha+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes Zygmund-H\"older of order $s$. We give necessary sufficient conditions for when there is $\mathscr{C}^{\beta+1}$ structure manifold, compatible with its structure, respect to are $\mathscr{C}^{\beta}_{\mathrm{loc}}$. This strengthens previous results first author dealt setting $\alpha>1$, $\beta>\max\{ \alpha, 2\}$.