I'm confused as to what your question is, since I answered the last question. Then you asked essentially the same question.
There is no general solution to this problem. In fact, it can be proved there can never be an analytical solution in general for an equation of degree 5 or higher. Why? because you can always resove the problem by multiplying by the largest negative exponent, which is valid as long as zeta is non-zero.
The result will be some general polynomial equation, but if the degree is higher than 4 (i.e., 5 or larger) then the Abel-Ruffini theorem comes into play, where it was proven that no algebraic solution will exist for that problem in terms of radicals. (Yes, there can be some relatively rare cases where a higher order polynomial does have a solution in terms of radicals. But those cases will be rare.)
So the very best you can do is multiply by some power of zeta. Colllect coefficients. Then call roots for any value of z you wish. There can be NO better solution. NO more general solution can ever exist, as long as the polynomial degree of your problem is 5 or greater.
Sorry, but asking the same question will not get you a better answer.