Linearization from complex Lie point transformations

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimension $d$, with $d\le 4$. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue in ${\mathbb{R}}^3$ of the linearizability criteria in ${\mathbb{R}}^2$.