This section lists all ode’s in the database of second order which Maple gave type as Lie symmetry with the symgen output of the transformation \(\xi (x,y),\eta (x,y)\) and the corresponding canonical transformation to the new coordinates \(r,s(r)\) (which is other places are written as \(X,Y(X)\). In otherwords, given the coordinates \(x,y(x)\), applying Lie transformation gives the coordinates \(r,s(r)\) in which the ode becomes quadrature and easily solved. Maple uses \(r,s(r)\) for \(X,Y(X)\). The transformation is given by \begin {align*} r(x,y) & =x+\epsilon \xi \left ( x,y\right ) \\ s(x,y) & =y+\epsilon \eta \left ( x,y\right ) \end {align*}
See this for more information about using Lie symmetry to solving odes.