Link to actual problem [2361] \[ \boxed {\left (y-y^{\prime } x \right )^{2}-{y^{\prime }}^{2}=1} \]
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 [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}-1}{x^{2}}, S \left (R \right ) &= -\operatorname {arctanh}\left (y\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {-1+x}\, \sqrt {1+x}}, S \left (R \right ) &= -\operatorname {arctanh}\left (x \right )\right ] \\ \end{align*}