  Simplifications in dealing with the equation of the center when the
short-period effects are removed from a zonal Hamiltonian are commonly
attributed to the elimination of parallactic terms. But this interpretation is
incorrect, and the simplifications rather stem from the removal of concomitant
long-period terms, an outcome that can also be achieved without need of
eliminating the parallax. To show that, a Lie transforms simplification is
invented that augments the exponents of the inverse of the radius and still
achieves analogous simplifications in handling the equation of the center to
those provided by the classical elimination of the parallax simplification. The
particular case in which the new transformation does not modify the exponents
of the parallactic terms of the original problem, leads to a new intermediary
of the main problem that, while keeping higher order effects of J2, is formally
analogous to Cid's first order radial intermediary.
