On motion around the collinear libration points in the elliptic R3BP with a bigger triaxial primary

•We locate the positions of collinear points both analytically and numerically for binary neutron stars.•We examine the stability of the collinear points.•Triaxiality affects the positions but not the nature of stability. They remain unstable. We investigate the motion of a test particle in the vicinity of a binary made of a triaxial primary and a spherical companion moving along elliptic orbits about their common barycenter in the neighborhood of collinear libration points. Their positions and stability are found to be affected by the triaxiality of the bigger primary, and by the semi-major axis and the eccentricity of the binary’s orbits as well. The analytic results obtained are applied to binary neutron stars consisting of a bigger triaxial primary and a spherical companion. A numerical analysis shows that the positions of the collinear points of PSR J1518+4904, PSR B1534+12, PSR B1914+16, and PSR B2127+11c are affected by the triaxiality of the bigger primary, and by the eccentricity of the orbits. The stability behavior however remains unchanged: the collinear points remain unstable in the Lyapunov sense.