This work investigates the qualitative dynamics of a massless particle around elongated bodies modeled with a linearly varying density function. We assume that the segment rotates uniformly with angular velocity \(\omega\) , with the fixed segment considered as a special case. For the rotating straight segment, we identify four equilibria: two collinear and two triangular. The primary novelty of our work is the detection of a pitchfork bifurcation, wherein, under slow and moderate rotation, the triangular equilibria shift toward and merge with one of the collinear points, while remaining distinct in fast rotations. This feature, previously unreported, is visualized through Hill’s regions and highlights how both rotation rate and asymmetry in mass distribution affect orbital dynamics, providing new insights into gravitational environments near irregularly shaped bodies. Additionally, we analyze the linear stability of the relative equilibria. Finally, for the fixed straight segment, we examine singular solutions, showing that any solution not defined globally over time leads the particle to eventually approach the segment at a distance of zero. In the one-dimensional problem, all singularities are observed to result from collisions.
