Solve x^2y+xy^2+y^2z+yz^2+{z}^2x+z{x}^2=2019

$${ x }^{ 2 } y+x { y }^{ 2 } + { y }^{ 2 } z+y { z }^{ 2 } + { z }^{ 2 } x+z { x }^{ 2 } =2019$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{y^{4}-4yz^{3}+8076y+z^{4}-4zy^{3}+8076z-6\left(yz\right)^{2}}-y^{2}-z^{2}}{2\left(y+z\right)}\text{; }x=-\frac{\sqrt{y^{4}-4yz^{3}+8076y+z^{4}-4zy^{3}+8076z-6\left(yz\right)^{2}}+y^{2}+z^{2}}{2\left(y+z\right)}\text{, }&y\neq -z\text{ and }y^{4}-6y^{2}z^{2}-4yz^{3}+8076y+z^{4}-4zy^{3}+8076z\geq 0\\x=\frac{2019}{2y^{2}}\text{, }&z=-y\text{ and }y\neq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\frac{\sqrt{x^{4}-4xz^{3}+8076x+z^{4}-4zx^{3}+8076z-6\left(xz\right)^{2}}-x^{2}-z^{2}}{2\left(x+z\right)}\text{; }y=-\frac{\sqrt{x^{4}-4xz^{3}+8076x+z^{4}-4zx^{3}+8076z-6\left(xz\right)^{2}}+x^{2}+z^{2}}{2\left(x+z\right)}\text{, }&x\neq -z\text{ and }x^{4}-6x^{2}z^{2}-4xz^{3}+8076x+z^{4}-4zx^{3}+8076z\geq 0\\y=\frac{2019}{2x^{2}}\text{, }&z=-x\text{ and }x\neq 0\end{matrix}\right.$$