This paper studies the motion of an infinitesimal mass around seven equilibrium points in the framework of the elliptic restricted three-body problem under the assumption that the primary of the system is a non-luminous, oblate spheroid and the secondary is luminous. A practical application of this case could be the study of the dynamical evolution of dust particles in orbits around a binary system with a dark degenerate primary and a secondary stellar companion. Conditional stability of the motion around the triangular points exists for 0?< μ?< μc , where μ is the mass ratio. The critical mass ratio value μc depends on the combined effect of radiation pressure, oblateness, eccentricity, and the semimajor axis of the elliptic orbits; an increase in any of these parameters has destabilizing results on the orbits of the test particles. The overall effect is therefore that the size of the region of stability decreases when the value of these parameters increases. The collinear points and the out-of-plane equilibrium points are found to be unstable for any combination of the parameters considered here. Further, a numerical exploration computing the positions of the triangular points and the critical mass ratio of two binaries RX?J0450.1-5856 and Nova Cen 1969 (Cen X-4) is given.
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