Link to actual problem [8654] \[ \boxed {\left (3 y^{3} x -4 y x +y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right )=0} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= {\mathrm e}^{-\frac {\ln \left (y^{2}-2\right )}{2}-2 \ln \left (y \right )}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= x \sqrt {y^{2}-2}\, y^{2}\right ] \\ \end{align*}