Singularities of 3D vector fields preserving the Martinet form

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