Generalizations of the McMillan map to $N$-body systems

The McMillan map is a well-known example of a rational integrable system for one particle in a two-dimensional phase space. An elegant recent paper presented a generalization of the McMillan map to an $N$-body system, for particles moving in $d$ space dimensions. This paper presents some alternative generalizations (also completely integrable) of the McMillan map to $N$-body systems. In all cases, the phase space is foliated by a biquadratic curve in the dynamical variables (and a set of suitably chosen angular momentum variables). It is also demonstrated that the constraints to generalize the McMillan map to $N$-body systems are not trivial.