We investigate the qualitative characteristics of a test particle attracted to an irregular elongated body, modeled as a non-homogeneous straight segment with a variable linear density. By deriving the potential function in closed form, we formulate the Hamiltonian equations of motion for this system. Our analysis reveals a family of periodic circular orbits parameterized by angular momentum. Additionally, we utilize the axial symmetry resulting from rotations around the segment's axis to consider the corresponding reduced system. This approach identifies several reduced-periodic orbits by analyzing appropriate Poincar & eacute; sections. These periodic orbits are then reconstructed into quasi-periodic orbits within the full dynamical system.