The motion of a zero-mass point under the action of gravitation toward a central body and a perturbing acceleration P is considered. The magnitude of P is taken to be small compared to the main acceleration due to the gravitation of the central body, and the components of the vector P are taken to be constant in a reference frame with its origin at the central body and its axes directed along the velocity vector, normal to the velocity vector in the plane of the osculating orbit, and along the binormal. The equations in the mean elements were obtained in an earlier study. The algorithm used to solve these equations is given in this study. This algorithm is analogous to one constructed earlier for the case when P is constant in a reference frame tied to the radius vector. The properties of the solutions are similar. The main difference is that, in the most important cases, the quadratures to which the solution reduces lead to non-elementary functions. However, they can be expressed as series in powers of the eccentricity e that converge for e < 1, and often also for e = 1.