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.