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We prove the integrability of the discretization of the Neumann system recently proposed by V. Adler., 9 pp., LaTeX

We demonstrate that in a certain gauge the Lax matrix of the Ruijsenaars-Schneider (RS) model has a quadratic r -matrix structure independent of the relativistic parameter of the model. In particular, the dynamical r -matrix object governing the hierarchy of Lax equations connected with the RS model, turns out to be identical with the r -matrix of the Calogero-Moser (CM) model. In the degenerate cases (hyperbolic and rational) this phenomenon is partly explained by a geometric derivation of Lax equations for arbitrary flows of both the RS and the CM hierarchies.

We analyse the property of phase volume preservation for the partitioned Runge-Kutta methods. A family of explicit methods known to be symplectic is demonstrated to possess the volume preservation property as well. Runge-Kutta-Nystrom methods are also considered.

A discretization of the peakons lattice is introduced, belonging to the same hierarchy as the continuous--time system. The construction examplifies the general scheme for integrable discretization of systems on Lie algebras with $r$--matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow is found. A variational (Lagrangian) formulation is also given., Comment: 14 pages, LaTeX

We introduce two new discretizations of the Toda lattice related to the qd algorithm. They are demonstrated to belong to the same hierarchy as the continuous-time system, and to exemplify the general scheme for symplectic maps on Lie algebras with r -matrix Poisson brackets. The initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for both maps.

We propose a family of symplectic standard-like maps, providing finite-difference approximations for all integrable nonlinear oscillators related to the classical series of symmetric spaces, introduced by Fordy et al. The Lax pair representations are found and the complete integrability is proved for all but the BDI maps. For the latter a large but incomplete set of integrals is found. For the AIII system also the r-matrix structure is presented.