Link to actual problem [8822] \[ \boxed {y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+y^{2}=-4 a^{2}+4 a x} \]
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 {2 a}{y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 a x}{y}+y\right ] \\ \left [R &= -\frac {4 x a -y^{2}}{x^{2}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -6 x a -x^{2}+y^{2}, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 x \left (2 a^{2}-3 x a +y^{2}\right )}{y}\right ] \\ \operatorname {FAIL} \\ \end{align*}