Applied Physics and Mathematics Annotation << Back ON APSIDAL PRECESSION OF NON-ELLIPTICAL OVAL ORBITS AS AN EXISTING SOLUTION TO THE NEWTONIAN TWO-BODY PROBLEM. PART 2 A.V. KUKUSHKIN Within the boundaries of the revisited classical two-body problem a linear ordinary differential equation of the second order is obtained relative to the relative orbit’s fluctuation function’s derivative of the eccentric anomaly if the orbit’s form is presented as a Keplerian ellipse rotating at a low angular velocity round the centre of masses of a binary star. A strict proof is presented that for weakly eccentric orbits this equation has an exact solution in the form of a uniformly convergent even Fourier series for the derivative and respectively in the form of an odd Fourier series for the fluctuating function itself. It is thus proved that the Newton two-body equation has exact solutions outside Kepler’s first law. Geometrical properties of the new orbits’ symmetries are analyzed to show that they satisfy all possible requirements which could possibly be imposed on them by physics. It is shown that at very low apsidal velocities such as observed in astronomy the new rotating orbits are so close to ellipses that it is impossible to notice their deviation from the ellipse. However, the very fact of their existence as a solution to the two-body problem, a key problem of celestial mechanics, provides a clue to the explanation and description of the otherwise inexplicable orbits found in eclipsing binary stars GG Ori, V974 Cyg, and BW Aqr. Keywords: relative apsidal speed, eclipsing binary stars, orbital function, rotating complex plane, uniform orbital precession, non-uniform orbital precession, relative orbit, Kepler’s first law. DOI: 10.25791/pfim.02.2022.1224 Pp. 10-29.