  In the late 80s - early 90s J. Moser and A. P. Veselov considered Lagrangian
discrete systems on Lie groups with additional symmetry conditions imposed on
Lagrangians. They observed that such systems are often integrable
time-discretizations of integrable Euler equations on these Lie groups. In
recent papers we studied Lagrangian discrete systems with additional symmetry
requirements on certain infinite-dimensional Lie groups. We will discuss some
interesting properties of these systems.
