  This paper explores the problem of analytically approximating the orbital
state for a subset of orbits in a rotating potential with oblateness and
ellipticity perturbations. This is done by isolating approximate differential
equations for the orbit radius and other elements. The conservation of the
Jacobi integral is used to make the problem solvable to first-order in the
perturbations. The solutions are characterized as small deviations from an
unperturbed circular orbit. The approximations are developed for near-circular
orbits with initial mean motion $n_{0}$ around a body with rotation rate $c$.
The approximations are shown to be valid for values of angular rate ratio
$\Gamma = c/n_{0} > 1$, with accuracy decreasing as $\Gamma \rightarrow 1$, and
singularities at and near $\Gamma = 1$. Extensions of the methodology to
eccentric orbits are discussed, with commentary on the challenges of obtaining
generally valid solutions for both near-circular and eccentric orbits.
