Link to actual problem [8825] \[ \boxed {y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}=-a \,x^{2}-\left (a -1\right ) b} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_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 &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {a x}{y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x, \underline {\hspace {1.25 ex}}\eta &= y -\frac {b}{y}\right ] \\ \left [R &= -\frac {-y^{2}+b}{x^{2}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 2 x^{2} a -x^{2}+y^{2}-2 b +\frac {2 b}{a}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x \left (x^{2} a -a \,y^{2}+2 y^{2}\right )}{y}\right ] \\ \operatorname {FAIL} \\ \end{align*}