Link to actual problem [6805] \[ \boxed {y^{\prime } \left (y^{\prime } x -y+k \right )=-a} \]
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 &= -\frac {4 x a +2 k y-y^{2}}{4 a}, S \left (R \right ) &= \frac {y}{2 a}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {-y+k}{\sqrt {x}}, S \left (R \right ) &= \frac {\ln \left (x \right )}{2}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {4 x a -k^{2}+2 k y-y^{2}}{\left (a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}\right ) a}, S \left (R \right ) &= \int _{}^{y}\frac {1}{-9 k^{2}-\frac {2 \left (-\frac {3 \left (4 x a -k^{2}+2 k y-y^{2}\right ) k^{2}}{a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}}-\frac {\left (4 x a -k^{2}+2 k y-y^{2}\right ) k \textit {\_a}}{a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}}+\sqrt {-\frac {\left (4 x a -k^{2}+2 k y-y^{2}\right ) \textit {\_a}^{2}}{a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}}-\frac {2 \left (4 x a -k^{2}+2 k y-y^{2}\right ) k \textit {\_a}}{a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}}+\frac {3 \left (4 x a -k^{2}+2 k y-y^{2}\right ) k^{2}}{a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}}+4}+2\right ) \left (a^{2} x^{2}-2 a \,k^{2} x +2 a k x y+k^{4}-2 k^{3} y+k^{2} y^{2}\right )}{4 x a -k^{2}+2 k y-y^{2}}+\textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \end{align*}