Integrable Time-Dependent Dynamical Systems: G eneralized Ermakov-Pinney and Emden-Fowler Equations

We consider the integrable time-dependent classical dynamics studied by Bartuccelli and Gentile (Phys Letts. A307 (2003) 274-280; Appl. Math. Lett. 26 (2013) 1026-1030) and show its power to compute the first integrals of the (general- ized) Ermakov-Pinney systems. A two component generalization of the Bartuccelli- Gentile equation is also given and its connection to Ermakov-Ray-Reid system and coupled Milne-Pinney equation has been illucidated. Finally, we demonstrate its application in other integrable ODEs, in particular, using the spirit of Bartuccelli- Gentile algorithm we compute the first integrals of the Emden-Fowler and describe the Lane-Emden type equations. A number of examples are given to illustrate the procedure.