Assume that $\dim \mathcal C(x_0)=n$ and that, on some neighbourhood of $x_0$, the distribution $\Delta_{n-1}$ has constant rank $n-1$ and is involutive. Then the system is locally static state-feedback linearisable near $x_0$: there exist an open neighbourhood $V \subset U$ of $x_0$, a diffeomorphism $\Phi:V\to \Phi(V)\subset\mathbb{R}^n$, and a smooth feedback transformation
with nonzero input coefficient such that, in coordinates $z=\Phi(x)$ and with new input $v\in\mathbb{R}$, the system becomes the controllable chain of integrators