Symmetries of second-order systems of ODEs and integrability

We present a method for finding a complete set of kth-order (ka parts per thousand yen2) differential invariants including bases of invariants corresponding to vector fields in three variables of four-dimensional real Lie algebras. As a consequence, we provide a complete list of second-order differential invariants and canonical forms for vector fields of four-dimensional Lie algebras and their admitted regular systems of two second-order ODEs. Moreover, we classify invariant representations of these canonical forms of ODEs into linear, partial linear, uncoupled, and partial uncoupled cases. In addition, we give an integration procedure for invariant representations of canonical forms for regular systems of two second-order ODEs admitting four-dimensional Lie algebras.