The normalG2 command receives two pairs of infinitesimals, and an indication of the dependent variable y(x), and returns a sequence of infinitesimals $\mathrm{Y1},\mathrm{Y2}$, each one of the form $\left[\mathrm{\xi }\,\mathrm{\eta }\right]$, such that $\mathrm{Y1}$ and $\mathrm{Y2}$ are built using linear combinations of X1 and X2, and $\left[\mathrm{Y1}\,\mathrm{Y2}\right]=\mathrm{Y1}$, where $\left[\mathrm{Y1}\,\mathrm{Y2}\right]$ is the commutator of the two infinitesimals.

This command presently accepts only point symmetries, and when the given $\mathrm{X1},\mathrm{X2}$ do not form a solvable algebra (the problem has no solution), the command returns FAIL.

This function is part of the DEtools package, and so it can be used in the form normalG2(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[normalG2](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{X1}≔\left[ax\,cy\right]$

$\mathrm{X1}≔\left[ax\,cy\right]$

$\mathrm{X2}≔\left[ax\,cy+y^2\right]$

$\mathrm{X2}≔\left[ax\,cy+y^2\right]$

$\mathrm{Xcommutator}\left(\mathrm{X1}\,\mathrm{X2}\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=c{y}^2\right]$

$Y≔\mathrm{normalG2}\left(\mathrm{X1}\,\mathrm{X2}\,y\left(x\right)\right)$

$Y≔\left[0\,-y^2\right],\left[-\frac{ax}{c}\,-\frac{cy+y^2}{c}\right]$

$\mathrm{Xcommutator}\left(Y\left[1\right]\,Y\left[2\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=-y^2\right]$