6. Find all real pairs $(x, y)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
x^{3}+9 x^{2} y=10, \\
y^{3}+x y^{2}=2
\end{array}\right.
$$

$(x, y)=$ $\qquad$
$=, 6 .\left(\frac{5+3 \sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}\right),\left(\frac{5-3 \sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}\right)$, $(1,1)$.
$$
\begin{array}{l}
\text { From }(x+3 y)^{3}=x^{3}+9 x^{2} y+27 x y^{2}+27 y^{3} \\
=10+2 \times 27=64 \\
\Rightarrow x+3 y=4 \Rightarrow x=4-3 y .
\end{array}
$$

Substituting the above equation into equation (2) yields
$$
\begin{array}{l}
y^{3}-2 y^{2}+1=0 \\
\Rightarrow y=1, \frac{1 \pm \sqrt{5}}{2} \\
\Rightarrow x=1, \frac{5 \mp 3 \sqrt{5}}{2} .
\end{array}
$$