Invariant Lagrangians and a new method of integration of nonlinear equations

Abstract. A method for solving the inverse variational problem for di®er-ential equations admitting a Lie group is presented. The method is used for de-termining invariant Lagrangians and integration of second-order nonlinear di®er-ential equations admitting two-dimensional non-commutative Lie algebras. The method of integration suggested here is quite di®erent from Lie's classical method of integration of second-order ordinary di®erential equations based on canonical forms of two-dimensional Lie algebras. The new method reveals existence and signi¯cance of one-parameter families of singular solutions to nonlinear equations of second order