The family of potentials admitting integration of the perturbed two-body problem in regular coordinates

An infinite system of potentials is presented that admits separation of regular variables in the perturbed two-body problem. The regular coordinates are constructed using a specially selected L-matrix. An explicit solution to the problem in the elliptical case is constructed. In the general case the solution is reduced to inversion of hyper-elliptic integrals. The cases of motion with and without constraints are considered. The results of numerical experiments are presented.