Suppose $P(x,y)$ is a polynomial with real coefficients. Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$ has the property that $y_0 = \overline{x_0}$ (i.e. they are conjugate), provided that the number of solutions is finite?
Notice that if $(x,y)$ solves the equation, then $(\overline{x},\overline{y})$ also solves the system, and hence $(\overline{y},\overline{x})$ does too. Thus, the set of solutions is symmetric w.r.t both reversing order of coordinates and conjugation of both coordinates.
A nice proof would be to just count the number of solutions of the form $(x,\overline{x})$, and just use Bezouts theorem to see that there can't be any more, but I have no success proving that there are sufficiently many of that form.