We subtract the equations in pairs, giving $(x-y)-(x^3-y^3)=0, (y-z)-(y^3-z^3)=0, (x-z)-(x^3-z^3)=0$. These are equivalent to $(x-y)(x^2+y^2+xy-1)=0, (y-z)(y^2+z^2+yz-1)=0, (x-z)(x^2+z^2+xz-1)=0$. Now casework on which term is $0$ for each one.

Very easy problem for A3

Here is the solution in $\mathbb{C}$. All permutations of $(1,0,0)$ $(1,1,-1)$ $(A_1,A_2,A_3)$ $(B_n,B_n,B_n)$ (for $n=1,2,3$) $(C_\pm,C_\pm,D_\mp)$ $(E_\pm,E_\pm,F_\mp)$ $A_1=\frac{1}{6} \left(\sqrt[3]{108-12 \sqrt{69}}+\sqrt[3]{108+12 \sqrt{69}}\right)$ $A_{2,3}=-\frac{1}{12} \left(\sqrt[3]{108-12 \sqrt{69}}+\sqrt[3]{108+12 \sqrt{69}}\right)\pm\frac{i \sqrt{3}}{12}\left(\sqrt[3]{108+12 \sqrt{69}}-\sqrt[3]{108-12 \sqrt{69}}\right)$ $B_1=\frac{1}{6} \left(\sqrt[3]{108+12 \sqrt{177}}-\sqrt[3]{-108+12 \sqrt{177}}\right)$ $B_{2,3}=-\frac{1}{12} \left(\sqrt[3]{108+12 \sqrt{177}}-\sqrt[3]{-108+12 \sqrt{177}}\right)\pm\frac{i\sqrt{3}}{12} \left(\sqrt[3]{108+12 \sqrt{177}}+\sqrt[3]{-108+12 \sqrt{177}}\right)$ $C_\pm=-\frac{1}{4}-\frac{1}{12} \sqrt{-39+6 \sqrt[3]{-404-12 i \sqrt{687}}+6 \sqrt[3]{-404+12 i \sqrt{687}}}\pm\frac{1}{12} \sqrt{-78-6 \sqrt[3]{-404-12 i \sqrt{687}}-6 \sqrt[3]{-404+12 i \sqrt{687}}-6 \sqrt{-87+6 \sqrt[3]{-9388-588 i \sqrt{687}}+6 \sqrt[3]{-9388+588 i \sqrt{687}}}}$ $D_\pm=-\frac{1}{6} \sqrt{6+3 \sqrt[3]{161-6 i \sqrt{687}}+3 \sqrt[3]{161+6 i \sqrt{687}}}\pm\frac{1}{6} \sqrt{12-3 \sqrt[3]{161-6 i \sqrt{687}}-3 \sqrt[3]{161+6 i \sqrt{687}}-6 \sqrt{-33+3 \sqrt[3]{-719-24 i \sqrt{687}}+3 \sqrt[3]{-719+24 i \sqrt{687}}}}$ $E_\pm=-\frac{1}{4}+\frac{1}{12} \sqrt{-39+6 \sqrt[3]{-404-12 i \sqrt{687}}+6 \sqrt[3]{-404+12 i \sqrt{687}}}\pm\frac{1}{12} \sqrt{-78-6 \sqrt[3]{-404-12 i \sqrt{687}}-6 \sqrt[3]{-404+12 i \sqrt{687}}+6 \sqrt{-87+6 \sqrt[3]{-9388-588 i \sqrt{687}}+6 \sqrt[3]{-9388+588 i \sqrt{687}}}}$ $F_\pm=\frac{1}{6} \sqrt{6+3 \sqrt[3]{161-6 i \sqrt{687}}+3 \sqrt[3]{161+6 i \sqrt{687}}}\pm\frac{1}{6} \sqrt{12-3 \sqrt[3]{161-6 i \sqrt{687}}-3 \sqrt[3]{161+6 i \sqrt{687}}+6 \sqrt{-33+3 \sqrt[3]{-719-24 i \sqrt{687}}+3 \sqrt[3]{-719+24 i \sqrt{687}}}}$