= Solution
To first order, the initial-phase correction in $M_0$ can be omitted inside a term already multiplied by $e$. Thus
$$
\frac ra=1-e\cos(nt-\varpi)+O(e^2),\qquad
\delta\lambda=(n-n_g)t+2e\sin(nt-\varpi)+2e\sin\varpi+O(e^2).
$$
For $|a-a_g|/a\ll1$ and $|\delta\lambda|\ll1$, expand the exact <rotating reference frame> coordinates. Products of the small radial and angular displacements are second order. This gives
$$
\boxed{\frac xa\simeq\frac{a-a_g}{a}-e\cos(nt-\varpi),\qquad
\frac ya\simeq(n-n_g)t+2e\sin(nt-\varpi)+2e\sin\varpi.}
$$
At $t=0$ the two sine terms cancel, as required. Equal <mean motion> gives a closed local <epicyclic motion> with radial-to-tangential amplitude ratio $1:2$; unequal <mean motion> adds along-track drift. The local expansion also requires $|(n-n_g)t|\ll1$, so small frequency mismatch alone does not justify it for arbitrarily large times.
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