This article examines the relationship between geometric Poisson brackets and integrable systems in flat Galilean and Minkowski spaces. First, moving frames are used to calculate differential invariants of curves and to write invariant evolution equations. The Galilean moving frame is the limit of the Minkowski one as c ! 1. Then, associated integrable evolutions and their bi-Hamiltonian structures are found, using the parallelism of Euclidean and Minkowski cases. The Galilean case is significant because its group is not semisimple, yet it can be considered as a limit of the (semisimple) Minkowski case. The Galilean integrable systems and Hamiltonian structures are compared to the c ! 1 limit of the Minkowski ones.