  It is proved that the members of the Riccati hierarchy, the so-called Riccati
chain equations, can be considered as particular cases of projective Riccati
equations, which greatly simplifies the study of the Riccati hierarchy. This
also allows us to characterize Riccati chain equations geometrically in terms
of the projective vector fields of a flat Riemannian metric and to easily
derive their associated superposition rules. Next, we establish necessary and
sufficient conditions under which it is possible to map second-order Riccati
chain equations into conformal Riccati equations through a local
diffeomorphism. This fact can be used to determine superposition rules for
particular higher-order Riccati chain equations which depend on fewer
particular solutions than in the general case. Therefore, we analyze the
properties of Euclidean, hyperbolic and projective vector fields on the plane
in detail. Finally, the use of contact transformations enables us to apply the
derived results to the study of certain integrable partial differential
equations, such as the Kaup-Kupershmidt and Sawada-Kotera equations.
