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

Singularities of 3d vector fields preserving the form of Martinet