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Problem Integrating a sequence of polynomials


These polynomials are x^n+y^n+z^n=0
 Weierstrass-Beauville parametrics.
The results are Finsler space like minimal surfaces.
The z coordinate comes out to be Riemannian surface "sheet like"
and Mathematica really doesn't like my approach
to the problem.

Here is my code so far after about three or four attempts:
Clear[x, y, z, g, f]
x[n_] = If[Mod[n, 2] == 1, -f*(g^n + 1), I*f*(g^n + 1)]
y[n_] = f*(g^n - 1)
Table[Solve[x[n]^n + y[n]^n + z^n == 0, z], {n, 1, 5}]
w = {-(g^5 + 1), (g^5 - 1), (1 + 10  g^10 + 5  g^20)^(1/5)}
ParametricPlot3D[w, {g, -1, 1}]
ParametricPlot3D[w, {g, -10, 10}]
a = Table[-I*
   If[Mod[n, 2] == 1,
    w = {x[n]/f,
      y[n]/f, (z/f) /. Solve[x[n]^n + y[n]^n + z^n == 0, z][[1]]},
    w = {x[n]/f,
      y[n]/f, (z/f) /.
       Solve[x[n]^n + y[n]^n + z^n == 0, z][[1]]}], {n, 1, 5}]
z = r*Exp[i*t]
b = a /. g -> z /. f -> 1
Clear[x]
c = Table[Re[NIntegrate[b[[n]], {t, 0, x}]], {n, 1, Length[b]}]
Table[ParametricPlot3D[c[[n]], {r, 0, 1}, {x, 0, 2*Pi}], {n, 1,
  Length[c]}]



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