= Solution
For $0<e<1$, differentiate the logarithm of the proposed <Poynting–Robertson drag invariant>:
$$
\frac{\dot C}{C}=\frac{\dot a}{a}-\frac{2e\dot e}{1-e^2}-\frac45\frac{\dot e}{e}.
$$
Substitution of the orbit-averaged <Poynting–Robertson drag> rates gives, over the common denominator $a^2(1-e^2)^{3/2}$, the numerator $A[-(2+3e^2)+5e^2+2(1-e^2)]=0$. Hence
$$
\boxed{C=a(1-e^2)e^{-4/5}=\text{constant}.}
$$
Equivalently, eliminating time gives $d\log a/de=(2/5)(2+3e^2)/[e(1-e^2)]$, whose integral is $\log a=(4/5)\log e-\log(1-e^2)+\text{constant}$. An exactly circular orbit remains circular; the expression for $C$ is singular at $e=0$ and is replaced by that separate solution.
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