Application of the λλ-symmetries approach and time independent integral of the modified Emden equation

In this paper we derive the time-independent integral for a nonlinear dissipative system, namely the modified Emden equation, from Lie point symmetries. We employ the recently introduced λλ-symmetries method [C. Muriel, J.L. Romero, First integrals, integrating factors and λλ-symmetries of second-order differential equations, J. Phys. A: Math. Theor. 42 (2009) 365207–365217] to complete this task. To begin with we recall Lie point symmetries of this system and derive λλ-symmetries from the vector fields. The knowledge of λλ-symmetries enables us to obtain integrating factors, integrals and the general solution for the linearizable case. While determining the integrating factor from the λλ-symmetry for the integrable case we find that this case splits up into three sub-cases. We then obtain the integrating factor and integral for these three sub-cases. The results agree with the ones reported in the literature and thereby give a group theoretical interpretation for the nonstandard time independent integrals exhibited by the system.