= Solution
Write $\epsilon=\mu^{1/3}$ and $\delta_j=(a_j-a_0)/a_0$. The leading <Keplerian shear> is
$$
\frac{n_j-n_0}{n_0}=-\frac32\delta_j+O(\delta_j^2,\mu).
$$
The distinction between total mass $m$ and the individual <planet>'s two-body central mass is order $\mu$, below the retained order $\epsilon$.
At one common reference epoch, the local <Kepler orbit> expansion takes the form
$$
\frac{x'_j}{a_0}-1\simeq\delta_j-e_j\cos(n_0t-\varpi_j),\qquad
\frac{y'_j}{a_0}\simeq-\frac32\delta_j n_0t+2e_j\sin(n_0t-\varpi_j)+\lambda_{j,0},
$$
where $\lambda_{j,0}$ is the initial <mean longitude> relative to the rotating reference ray. Comparing its sine and cosine coefficients with the <free solution of Hill equations> gives
$$
D_{1j}=-\frac{e_j}{\epsilon}\cos\varpi_j,\qquad
D_{2j}=-\frac{e_j}{\epsilon}\sin\varpi_j,\qquad
D_{3j}=\frac{\delta_j}{\epsilon},\qquad
D_{4j}=\frac{\lambda_{j,0}}\epsilon.
$$
The required <orbital elements from Hill coordinates> are therefore
$$
\boxed{a_j=a_0(1+\epsilon D_{3j}),\qquad
e_j=\epsilon\sqrt{D_{1j}^2+D_{2j}^2},\qquad
\varpi_j=\operatorname{atan2}(-D_{2j},-D_{1j})\pmod{2\pi}.}
$$
These equalities have the first-order accuracy of the local approximation. If $e_j=0$, the <longitude of periapsis> is undefined. The combination $D_{4j}+2D_{2j}$ fixes the initial true longitude divided by $\epsilon$. The special initial alignment imposed in part v would require $D_{4j}=-2D_{2j}$, but the general two-planet solution need not impose it.
Substitution also verifies $\ddot\xi_j-2n_0\dot\eta_j-3n_0^2\xi_j=0$ and $\ddot\eta_j+2n_0\dot\xi_j=0$: these are the unforced <Hill equations>. Close encounters add mutual-gravity forcing and can change the constants. Replacing $n_j$ by $n_0$ in the oscillation is consistent at leading order over local orbital times; accumulated phase differences must be retained when following much longer evolution.
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