Singular invariant structures for Lie algebras admitted by a system of second-order ODEs

Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.