Classical and quantum integrable systems with boundaries

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