KöMaL 2024 C1831

Solve the following system of equations on the set of real numbers: $$\begin{gather*} 2x^3-3x^2y+2xy^2-y^3+1=0,\tag*{(1)}\\ x^3+2x^2y-xy^2-2y^3+3=0.\tag*{(2)} \end{gather*}$$