  We study the local structure of vector fields on $\mathbb{R}^3$ which
preserve the Martinet $1$-form $\alpha=(1+x)dy\pm zdz$. We present the
classification of their singularities, up to diffeomorphisms preserving the
form $\alpha$, as well as their transversal unfoldings. We are thus able to
provide a fairly complete list of the bifurcations such vector fields undergo.
