  We consider the hierarchy of higher-order Riccati equations and establish
their connection with the Gambier equation. Moreover we investigate the
relation of equations of the Gambier family to other nonlinear differential
systems. In particular we explore their connection to the generalized
Ermakov-Pinney and Milne-Pinney equations. In addition we investigate the
consequence of introducing Okamoto's folding transformation which maps the
reduced Gambier equation to a Li\'enard type equation. Finally the conjugate
Hamiltonian aspects of certain equations belonging to this family and their
connection with superintegrability are explored.

Exactly Solvable and Integrable Systems;Classical Analysis and ODEs