Link to actual problem [15121] \[ \boxed {\left (1+{y^{\prime }}^{2}\right ) y^{2}-4 y y^{\prime }=4 x} \]
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}{y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 2+x, \underline {\hspace {1.25 ex}}\eta &= y -\frac {2 x}{y}\right ] \\ \left [R &= -\frac {4-y^{2}+4 x}{x^{2}+4 x +4}, S \left (R \right ) &= \ln \left (2+x \right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2} x^{2}-\frac {1}{2} y^{2}+2+3 x, \underline {\hspace {1.25 ex}}\eta &= -\frac {x \left (-y^{2}+3 x +2\right )}{y}\right ] \\ \operatorname {FAIL} \\ \end{align*}