I'm trying to solve the equation
in the ring of complex polynomials; indeed, I don't know if I can find all solutions, prove there aren't non constant solutions or exhibit one.
One should certainly add a condition such as $\gcd(A,B,C,D)=1$ in order to avoid other kind of trivial solutions like $(A,B,-A,-B)$.
I tried to use the Mason-Stothers theorem, since it may be used to show Fermat's Last Theorem for polynomials or Catalan's conjecture for rational functions, but I could not get a strong enough condition on degrees to get a solution or a contradiction.
May anybody help me?