Solution of System of Bivariate Cubic Polynomials

Question

Apart from numerical solutions, is there a method to find the real roots of $X$ and $Y$ for this system of nonlinear equations ?

$X^3 -3 X Y^2 + b_1 X - b_2 Y + c_1=0$

$Y^3 -3 X^2 Y - b_1 Y - b_2 X - c_2=0$

where $b_i$ and $c_i$ are real constants.

Answer 1

take $Z = X + i Y,$ then $B= b_1 + i b_2$ and $C = c_i + i c_2.$ Then $$ Z^3 + BX + C = 0. $$ Cardano's method applies, as this is a depressed cubic already, no $Z^2.$

I would think that some choices of $B,C$ lead to nice things, but more commonly Cardano's leads to taking cube roots of complex numbers. Try some examples.