Link to actual problem [8948] \[ \boxed {y^{\prime }-\frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 &= \frac {y}{x}, \underline {\hspace {1.25 ex}}\eta &= a^{2}-1\right ] \\ \left [R &= -a^{2} x^{2}+y^{2}+x^{2}, S \left (R \right ) &= \frac {y}{\left (a -1\right ) \left (1+a \right )}\right ] \\ \end{align*}