  The construction of an analytic orbit theory that takes into account the main
effects of the Geopotential is notably simplified when splitting the removal of
periodic effects in several stages. Conversely, this splitting of the
analytical solution into several transformations reduces the evaluation
efficiency for dense ephemeris output. However, the advantage is twofold when
the different parts of the mean-to-osculating transformation are composed into
a single transformation. To show that, Brouwer's solution is extended to the
second order of the zonal harmonic of the second degree by the sequential
elimination of short- and long-period terms. Then, the generating functions of
the different transformations are composed into a single one, from which a
single mean-to-osculating transformation is derived. The new, unique
transformation notably speeds up the evaluation process, commonly improving
evaluation efficiency by at least one third with respect to the customary
decomposition of the analytical solution into three different parts.
