Link to actual problem [4223] \[ \boxed {2 y^{2} {y^{\prime }}^{2}+2 y y^{\prime } x +y^{2}=-x^{2}+1} \]
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 {1}{y}\right ] \\ \left [R &= \frac {y^{2}-1}{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 {x}{2 y}\right ] \\ \left [R &= \frac {x^{2}}{2}+y^{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {2 y^{2}}{3}-\frac {4}{3}, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{3}}{3 y}+x y\right ] \\ \operatorname {FAIL} \\ \end{align*}