  We consider a hierarchy of classical Liouville completely integrable models
sharing the same (linear) $r$--matrix structure obtained through an $N$--th
jet--extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of
the present paper is the study of the integrable model corresponding to N=3,
since the case N=2 has been considered by the authors in separate papers, both
in the one--body case (Lagrange top) and in the $n$--body one (Lagrange chain).
We now obtain a rigid body associated with a Lie--Poisson algebra which is an
extension of the Lie--Poisson structure for the two--field top, thus breaking
its semidirect product structure. In the second part of the paper we construct
an integrable discretization of a suitable continuous Hamiltonian flow for the
system. The map is constructed following the theory of B\"acklund
transformations for finite--dimensional integrable systems developed by V.B.
Kuznetsov and E.K. Sklyanin.
