57. (8th grade) Solve the system of equations $\left\{\begin{array}{l}\frac{x^{2}}{y}+\frac{y^{2}}{x}=12 \\ \frac{1}{x}+\frac{1}{y}=\frac{1}{3}\end{array}\right.$.
57. Given $x \neq 0, y \neq 0$, the system is equivalent to the system $\left\{\begin{array}{l}x^{3}+y^{3}=12 x y \\ 3(x+y)=x y .\end{array}\right.$

If $x+y=v, \quad x y=3 v$, then we get the equation $v^{2}-9 v-36=0$.

From which $v_{1}=12, v_{2}=-3$ or $x_{1}=y_{1}=6, \quad x_{2}=\frac{-3 \pm 3 \sqrt{5}}{2}, y_{2}=\frac{-3 \pm 3 \sqrt{5}}{2}$.