  Lattice systems with certain Lie algebraic or quantum Lie algebraic
symmetries are constructed. These symmetric models give rise to series of
integrable systems. As examples the $A_n$-symmetric chain models and the
SU(2)-invariant ladder models are investigated. It is shown that corresponding
to these $A_n$-symmetric chain models and SU(2)-invariant ladder models there
are exactly solvable stationary discrete-time (resp. continuous-time) Markov
chains with transition matrices (resp. intensity matrices) having spectra which
coincide with the ones of the corresponding integrable models.
