Linearization of systems of four second-order ordinary differential equations

In this paper we provide invariant linearizability criteria for a class of systems of four second-order ordinary differential equations in terms of a set of 30 constraint equations on the coefficients of all derivative terms. The linearization criteria are derived by the analytic continuation of the geometric approach of projection of two-dimensional systems of cubically semi-linear second-order differential equations. Furthermore, the canonical form of such systems is also established. Numerous examples are presented that show how to linearize nonlinear systems to the free particle Newtonian systems with a maximally symmetric Lie algebra relative to \(sl(6, \Re)\) of dimension 35.