Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation

There are seven equivalence classes of second-order ordinary differential equations possessing only three Lie point symmetries and hence not linearisible by means of a point transformation. We examine the representatives of these classes for linearisibility by means of other types of transformation. In particular we compare the potential for linearisibility and the possession of the Painlevé property. The complete symmetry group is realised in the standard algebra for each of the equivalence classes.