  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 literature.
