Link to actual problem [4158] \[ \boxed {\left (a^{2}+x^{2}\right ) {y^{\prime }}^{2}-2 y y^{\prime } x +y^{2}=-b} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
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 &= -\frac {a^{2} y}{b}, \underline {\hspace {1.25 ex}}\eta &= x\right ] \\ \left [R &= \frac {y^{2} a^{2}+b \,x^{2}}{a^{2}}, S \left (R \right ) &= -\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {y^{2} a^{2}}}\right )}{a}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}+b}{x^{2}}, S \left (R \right ) &= \frac {\arctan \left (\frac {y}{\sqrt {b}}\right )}{\sqrt {b}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {a^{2}+x^{2}}}, S \left (R \right ) &= \frac {\arctan \left (\frac {x}{a}\right )}{a}\right ] \\ \end{align*}