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We introduce a method for finding general solutions of third-order nonlineardifferential equations by extending the modified Prelle-Singer method. Wedescribe a procedure to deduce all the integrals of motion associated with thegiven equation so that the general solution follows straightforwardly fromthese integrals. The method is illustrated with several examples. Further, wepropose a powerful method of identifying linearizing transformations. Theproposed method not only unifies all the known linearizing transformationssystematically but also introduces a new and generalized linearizingtransformation (GLT). In addition to the above, we provide an algorithm toinvert the nonlocal linearizing transformation. Through this procedure thegeneral solution for the original nonlinear equation can be obtained from thesolution of the linear ordinary differential equation.Comment: Submitted to Proceedings of the Royal Society London Series A, 21  page

On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations