Link to actual problem [8824] \[ \boxed {y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}=x^{2}-a} \]
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 {x}{y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x, \underline {\hspace {1.25 ex}}\eta &= y +\frac {a}{2 y}\right ] \\ \left [R &= \frac {2 y^{2}+a}{2 x^{2}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -3 x^{2}+y^{2}+2 a, \underline {\hspace {1.25 ex}}\eta &= \frac {x \left (x^{2}-3 y^{2}\right )}{y}\right ] \\ \operatorname {FAIL} \\ \end{align*}