  In this paper we present an overview of the connection between completely
integrable systems and the background geometry of the flow. This relation is
better seen when using a group-based concept of moving frame introduced by Fels
and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The
paper discusses the close connection between different types of geometries and
the type of equations they realize. In particular, we describe the direct
relation between symmetric spaces and equations of KdV-type, and the possible
geometric origins of this connection.
