Link to actual problem [4493] \[ \boxed {2 y x +\left (x^{2}+y^{2}+a \right ) y^{\prime }=0} \]
type detected by program
{"exact"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{x^{2}+y^{2}+a}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} y+\frac {y^{3}}{3}+a y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (3 x^{2}+y^{2}+3 a \right )}{x^{2}+y^{2}+a}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (y \left (3 x^{2}+y^{2}+3 a \right )\right )}{3}\right ] \\ \end{align*}