### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 026, 23 pages      arXiv:1303.3434      https://doi.org/10.3842/SIGMA.2013.026
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

### A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

José F. Cariñena a, Partha Guha b and Javier de Lucas c
a) Department of Theoretical Physics and IUMA, University of Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain
b) S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata - 700.098, India
c) Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, Wóy-cickiego 1/3, 01-938, Warsaw, Poland

Received September 26, 2012, in final form March 14, 2013; Published online March 26, 2013

Abstract
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.

Key words: Lie system; Kummer-Schwarz equation; Milne-Pinney equation; quasi-Lie scheme; quasi-Lie system; second-order Gambier equation; second-order Riccati equation; superposition rule.

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