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(This is a redux of this.)

Why doesn't this work? I have a vector field F1 whose div is 0 with a known vector potential function A2, and I try to get DSolve to solve the differential equation for A2 as a "new" solution A1. Thanks in advance (TIA).

A2[q3_, q4_, q5_] = {q4/(q3^2 + q4^2 + q5^2), 
  q5/(q3^2 + q4^2 + q5^2), q3/(q3^2 + q4^2 + q5^2)}
F1[q3_, q4_, q5_] = Simplify[Curl[A2[q3, q4, q5], {q3, q4, q5}]]
Simplify[Div[F1[q3, q4, q5], {q3, q4, q5}]]
DSolve[Thread[
  Curl[{A11[q3, q4, q5], A12[q3, q4, q5], A13[q3, q4, q5]}, {q3, q4, 
     q5}] == {F1[q3, q4, q5][[1]], F1[q3, q4, q5][[2]], 
    F1[q3, q4, q5][[3]]}], {A11[q3, q4, q5], A12[q3, q4, q5], 
  A13[q3, q4, q5]}, {q3, q4, q5}]
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