Classical and quantum integrable systems with boundaries

We study two-dimensional classically integrable field theory with independe nt boundary conditions at each end, and obtain three possible generating fu nctions for integrals of motion when this model is an ultralocal one. Class ically integrable boundary conditions can be found when solving boundary K/- equations. In the quantum case, we also find that the unitarity conditio n of the quantum R-matrix is sufficient to construct commutative quantities with boundaries, and its reflection equations are obtained.