Applied Physics and Mathematics Annotation << Back ON EXISTENCE OF NEW POST-NEWTONIAN SOLUTIONS OF THE CLASSICAL TWO-BODY PROBLEM. PART 1 A.V. KUKUSHKIN Previously, when updating the formulation of the classical two-body problem, the author obtained an ordinary second-order linear differential equation with periodic coefficients with respect to the lambda function, which is the derivative of the ‘main disturber’ beta function with respect to the eccentric anomaly η (determined for the relative orbit), describing the perturbation of the reference ellipse selected to be the relative orbit rotating at a low angular velocity around the center of inertia of the binary system. At the previous stage, this equation was simplified by the author for the case of vanishingly small eccentricity values e of the reference ellipse, e << 1, which then allowed the author to integrate the equation and obtain a solution for the lambda function in the form of a convergent even trigonometric Fourier series. In this article, the eccentricity restriction is removed and the integral of this equation is found in the form of a similar Fourier series. It is shown that due to the above the differential operator forming this equation acquires the ability to translate said equation into an either trigonometric or power-law equation, with subsequent transition from those to an equivalent infinite-order system of inhomogeneous linear algebraic equations. It has been proven that it is in the latter case the matrix corresponding to the system of linear equations as composed of the coefficients of even trigonometric Fourier series of the lambda function is nondegenerate and well-conditioned. Evidence is given that the latter ensures uniform convergence of the odd Fourier series for the beta function along the entire η set: [– ∞, + ∞]. This means that for any e taken from the interval (0, 1) there is a solution of the Newtonian equation of motion for the two-body problem for the case of point masses performing complex motion with a very low translational angular velocity around a common center of inertia. The resulting new solution is a generalization of the old solution for elliptic orbits. This success achieved after more than three hundred years since Newton had obtained the old solution, was predetermined thanks to the analysis of Newton's equation of motion, which led the author to the inevitable conclusion that the shape of the rotating relative orbits cannot be elliptical. Keywords: two-body problem, eclipsing binary stars, orbital function, rotating complex plane, Chebyshev polynomials, relative apsidal velocity. DOI: 10.25791/pfi m.01.2024.1289 Pp. 42-59.