A pair of finite-dimensional integrable systems possessing the common non-dynamical r-matrix

In two recent papers [Phys. Lett. A 235 (1997) 35; J. Math. Phys. 39 (1998) 3271], some interesting conclusions were proposed that four pairs of different finite-dimensional integrable systems, which are the discrete Toda symplectic map and continuous constrained c-KdV system, AKNS and Dirac system, Harry–Dym and Heisenberg spin chain system, and Geng and Qiao system, respectively, possess the common non-dynamical r-matrix. A natural problem is whether there also exist any other pairs like them. In this paper, it is shown that the fifth pair of different finite-dimensional integrable systems of Kaup–Newell (KN) hierarchy and Yan hierarchy also share the same non-dynamical r-matrix being independent from the dynamical variables. The two finite-dimensional integrable systems possess Lax representation and can be shown to be Liouville integrable by using the r-matrix.