Suppose that we have a rational function of the form $$\displaystyle f(x) = \frac{P(x)}{Q(x)},$$ where $P,Q$ are polynomials of equal degree with rational coefficients, for which there exists two complex numbers $x_1, x_2$ such that $$\displaystyle x_1 = f(x_2)$$ and $$\displaystyle x_2 = f(x_1).$$ Is there a relatively simple, non-messy way to do this? I would also accept clear, dummy-proof computer algebra solutions. The out put should be in terms of the coefficients of $P,Q$.
Thanks for any help provided.