The study of planar free curves is a very active area of research, but a structural study of such a class ismissing. We give a complete classification of the possible generators of the Jacobian syzygy module of a plane free curve underthe assumption that one of them is linear. Specifically, we prove that, up to similarities, there are two possible forms for theHilbert–Burch matrix. Our strategy relies on a translation of the problem into the accurate study of the geometry of maximalsegments of a suitable triangle with integer points. Following this description, we are able to determine explicitly the equationsof free curves and the associated Hilbert–Burch matrices.
