On different integrable systems sharing the same nondynamical $r$-matrix

In a recent paper [Zhijun Qiao and Ruguang Zhou, Phys. Lett. A 235, 35 (1997)], the amazing fact was reported that a discrete and a continuous integrable system share the same $r$-matrix with the interesting property of being nondynamical. Now, we present three further pairs of different continuous integrable systems sharing the same $r$-matrix again being nondynamical. The first pair is the finite-dimensional constrained system (FDCS) of the famous AKNS hierarchy and the Dirac hierarchy; the second pair is the FDCS of the well-known geodesic flows on the ellipsoid and the Heisenberg spin chain hierarchy; and the third pair is the FDCS of one hierarchy studied by Xianguo Geng [Phys. Lett. A 162, 375 (1992)] and another hierarchy proposed by Zhijun Qiao [Phys. Lett. A 192, 316 (1994)]. All those FDCSs possess Lax representations and from the viewpoint of $r$-matrix can be shown to be completely integrable in Liouville’s sense.