  In this paper we consider affine Toda systems defined on the half-plane and
study the issue of integrability, i.e. the construction of higher-spin
conserved currents in the presence of a boundary perturbation. First at the
classical level we formulate the problem within a Lax pair approach which
allows to determine the general structure of the boundary perturbation
compatible with integrability. Then we analyze the situation at the quantum
level and compute corrections to the classical conservation laws in specific
examples. We find that, except for the sinh-Gordon model, the existence of
quantum conserved currents requires a finite renormalization of the boundary
potential.
