Canonical forms for systems of two second-order ordinary differential equations

We obtain non-similar classes of realizations for real three- and four-dimensional Lie algebras in the space of vector fields in three variables. This is applied to the classification and integration of systems of two second-order ordinary differential equations (ODEs) admitting four-dimensional symmetry Lie algebras. Thus we obtain an analogue of Lie's method of integrating scalar second-order ODEs admitting two-dimensional symmetry Lie algebras for systems of two second-order ODEs. Applications to physical problems are presented.