  This paper, in which we develop ideas introduced in \cite{MR}, focuses on
\emph{reduction methods} (basically, group actions or, more generally,
simmetries) for the bienergy. This type of techniques enable us to produce
examples of critical points of the bienergy by reducing the study of the
relevant fourth order PDE's system to ODE's. In particular, we shall study
rotationally symmetric biharmonic conformal diffeomorphisms between
\emph{models}. Next, we will adapt the reduction method to study an ample class
of $G-$invariant immersions into the Euclidean space. At present, the known
instances in these contexts are far from reaching the depth and variety of
their companions which have provided fundamental solutions to classical
problems in the theories of harmonic maps and minimal immersions. However, we
think that these examples represent an important starting point which can
inspire further research on biharmonicity. In this order of ideas, we end this
paper with a discussion of some open problems and possible directions for
further developments.
