Link to actual problem [4230] \[ \boxed {\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+y^{2} a=b \,x^{2}+a b} \]
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 &= 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 &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {b x}{\left (-b +a \right ) y}\right ] \\ \left [R &= \frac {a y^{2}-b \,x^{2}-b y^{2}}{-b +a}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {x^{2} a -a \,y^{2}+b \,x^{2}+b \,y^{2}+2 a^{2}-2 a b}{b}, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 y x a}{b}+\frac {x^{3}}{y}+x y\right ] \\ \operatorname {FAIL} \\ \end{align*}