A set of orbital elements to fully represent the zonal harmonics around an oblate celestial body

The candidate reproduces the paper’s construction: (i) a Sundman time change dθ/dt = pθ/r² that, together with α = pθ/r − μ/pθ, s = sinφ, γ = (pφ/pθ)cosφ, and node variables (β, ξ), yields a fully linear Kepler system with two harmonic-oscillator pairs and two constants of motion (matching the paper’s Eq. (26) linear system) ; (ii) for J2, the non-polynomial system (paper’s Eq. (56)) is converted into a polynomial ODE system by introducing Iθ = 1/pθ and ξ (paper’s Eqs. (57)–(62)), and the candidate’s right-hand sides coincide with those in the paper ; and (iii) for general zonals, the same strategy produces the paper’s polynomial system (paper’s Eq. (71)) . The only substantive discrepancy is a sign typo in the paper’s early identity for pθ (paper’s Eq. (6)), where the correct relation is pθ² = pφ² + pλ²/cos²φ; the candidate explicitly adopts the correct identity, which is also the one implicitly used later in the paper to recover dγ/dθ = −s (paper’s Eq. (25)) . Aside from such minor typographical slips, both presentations are mathematically consistent and essentially the same proof.