  In this paper we continue to study equivariant pencil liftings and
differential operators on the algebra of densities. We emphasize the role that
the geometry of the extended manifold plays. Firstly we consider basic
examples. We give a projective line of diff($M$)-equivariant pencil liftings
for first order operators, and the canonical second order self-adjoint lifting.
Secondly we study pencil liftings equivariant with respect to volume preserving
transformations. This helps to understand the role of self-adjointness for the
canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil
lifting which is derived from the full symbol calculus of projective
quantisation. We use the DLO-pencil lifting to describe all regular
proj-equivariant pencil liftings. In particular the comparison of these pencils
with the canonical pencil for second order operators leads to objects related
to the Schwarzian. Within this paper the question of whether the pencil lifting
factors through a full symbol map naturally arises.
