 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 $0=iA{z}_f+i{A}_1{z}_1$. Since $A=-A_{1}f({\alpha }_1)$,This assumes $A\ne 0$. If an external process made the planet precess at frequency $g$, the particular solution would instead be $z_f=A_1{z}_1/(g-A)$; the stated formula relies on the isolated single-planet case.