Linearization criteria for a system of two second-order ordinary differential equations

For a system of second-order ordinary differential equations conditions of linearizability to the form x '' = 0 are well known. However, an arbitrary linear system need not be equivalent via an invertible point transformation to this simple form. We provide the criteria for a system of two second-order equations to be mapped to the linear system of the general form. Necessary and sufficient conditions for linearization by means of a point transformation are given in terms of coefficients of the system. These results are illustrated with a number of examples.