The true and eccentric anomaly parametrizations of the perturbed Kepler motion

The true and eccentric anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits, by considering perturbations of the radial part of the kinetic energy in a form of a series of negative powers of the orbital radius. A toolchest of methods for averaging observables as functions of the energy $E$ and angular momentum $L$ is developed. A broad range of systems governed by the generic Brumberg force and recent applications in the theory of gravitational radiation involve integrals of these functions over a period of motion. These integrals are evaluated by using the residue theorem. In the course of this work two important questions emerge: (1) When does the true and eccentric anomaly parameter exist? (2) Under what circumstances and why are the poles in the origin? The purpose of this paper is to find the answer to these queries.