  Birational toggling on Gelfand-Tsetlin patterns appeared first in the study
of geometric crystals and geometric Robinson-Schensted-Knuth correspondence.
Based on these birational toggle relations, Einstein and Propp introduced a
discrete dynamical system called birational rowmotion associated with a
partially ordered set. We generalize birational rowmotion to the class of
arbitrary strongly connected directed graphs, calling the resulting discrete
dynamical system the $R$-system. We study its integrability from the points of
view of singularity confinement and algebraic entropy. We show that in many
cases, singularity confinement in an $R$-system reduces to the Laurent
phenomenon either in a cluster algebra, or in a Laurent phenomenon algebra, or
beyond both of those generalities, giving rise to many new sequences with the
Laurent property possessing rich groups of symmetries. Some special cases of
$R$-systems reduce to Somos and Gale-Robinson sequences.
