  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.
