ON COMPUTING LINEARIZING COORDINATES FROM SYMMETRY ALGEBRA

A characterization of the symmetry algebra of the $N$th-order ordinary
differential equations (ODEs) with maximal symmetry and all third-order
linearizable ODEs is given. This is used to show that such an algebra
$\mathfrak{g}$ determines – up to a point
transformation – only one linear equation whose symmetry algebra is
$\mathfrak{g}$ and an algorithmic procedure is given
to find the linearizing coordinates. The procedure is illustrated by
several examples from the literature.