[ Irrational Equations [ Symmetric Systems. Involutory Transformations ]

Solve the system of equations

$$
\left\{\begin{array}{l}
\left(x^{2}+x y+y^{2}\right) \sqrt{x^{2}+y^{2}}=185 \\
\left(x^{2}-x y+y^{2}\right) \sqrt{x^{2}+y^{2}}=65
\end{array}\right.
$$
By adding the given equations, we get $\left(x^{2}+y^{2}\right) \sqrt{x^{2}+y^{2}}=125$. From this, $x^{2}+y^{2}=5$. Substituting into the first equation, we get $x y=12$. Therefore, $x^{2}+y^{2}=25, x^{2} y^{2}=144$. By Vieta's theorem $\left\{x^{2}, y^{2}\right\}=\{9,16\}$. Considering that the signs of $x$ and $y$ are the same, we get the answer.

## Answer

$\{3,4\},\{-3,-4\}$.