Complete group classification of systems of two linear second-order ordinary differential equations

•A complete group classification of the general case of linear systems of two second-order ordinary differential equations is given.•Four cases of linear systems of equations with inconstant coefficients are obtained.•A preliminary study of systems of two second-order nonlinear equations of the form y″=F(x,y,z),z″=G(x,y,z) is given. We give a complete group classification of the general case of linear systems of two second-order ordinary differential equations excluding the case of systems which are studied in the literature. This paper gives the initial step in the study of nonlinear systems of two second-order ordinary differential equations. It can also be extended to systems of equations with more than two equations. Furthermore the complete group classification of a system of two linear second-order ordinary differential equations is done. Four cases of linear systems of equations with inconstant coefficients are obtained.