Link to actual problem [5884] \[ \boxed {\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\cos \left (\theta \right ) \phi \sin \left (\theta \right )=\frac {\cos \left (2 \theta \right )}{2}+1} \]
type detected by program
{"riccati"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {2}{-1+\cos \left (2 \theta \right )}\right ] \\ \\ \end{align*}