  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.
