= Solution
For one isolated perturbing <planet>, its <Kepler orbit> has fixed <longitude of periapsis>, so $z_1$ is constant and its secular forcing frequency is zero. A constant <forced eccentricity> satisfies $0=iAz_f+iA_1z_1$. Since $A=-A_1f(\alpha_1)$,
$$
\boxed{z_f=-\frac{A_1}{A}z_1=\frac{z_1}{f(\alpha_1)}.}
$$
This assumes $A\ne0$. If an external process made the <planet> precess at frequency $g$, the particular solution would instead be $z_f=A_1z_1/(g-A)$; the stated formula relies on the isolated single-planet case.
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