Link to actual problem [4210] \[ \boxed {y^{2} {y^{\prime }}^{2}+2 y y^{\prime } x -y^{2}=-a} \]
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 [R &= \frac {-x^{2}-y^{2}+a}{2 y^{4}}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\sqrt {-\frac {\left (-x^{2}-y^{2}+a \right ) \textit {\_a}^{4}}{y^{4}}-\textit {\_a}^{2}+a}\, \textit {\_a}}d \textit {\_a}\right ] \\ \end{align*}