Matteo Petrera

We give an overview of the integrability of the Hirota–Kimura discretizationmethod applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota–Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.

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 Bäcklund transformations for finite-dimensional integrable systems developed by V.B. Kuznetsov and E.K. Sklyanin.