Linearizability criteria for systems of two cubically semi-linear second order ordinary differential equations (ODEs) were obtained by geometric means using real symmetry analysis (RSA). Separately, complex symmetry analysis (CSA) was developed to provide means to discuss systems of two ODEs. It was shown that CSA provides a class of linearizable systems of two cubically semi-linear ODEs. Linearizability criteria for this class were also developed. It is proved that the two classes of linearizable systems of two ODEs, provided by CSA and RSA, are inequivalent under point transformations.

Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.

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 $d\leq 4$. 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 $\R^{3}$ of the linearizability criteria in $\R^2$.

We obtain equivalence transformations for general linear third order ordinary differential equation (ODE) using Lie infinitesimal method. These equivalence transformations are then employed to deduce associated invariants. Derived invariants are used to reduce the linear third order ODEs with variable coefficients to simpler equations of this family with constant coefficients. It is shown that these reductions help in identifying the canonical forms of the linear third order ODEs with 4, 5, and 7-dimensional Lie point symmetry algebras. Though this algebraic classification for linear third order ODEs has already been presented, here differential invariants are shown to provide an alternate procedure to recover the same. We provide MAPLE codes used to derive the said invariants.