  We investigate integrable boundary conditions (BCs) for the principal chiral
model on the half-line, and rational solutions of the boundary Yang-Baxter
equation (BYBE). In each case we find a connection with (type I, Riemannian,
globally) symmetric spaces G/H: there is a class of integrable BCs in which the
boundary field is restricted to lie in a coset of H; these BCs are parametrized
by G/H x G/H; there are rational solutions of the BYBE in the defining
representations of all classical G parametrized by G/H; and using these we
propose boundary S-matrices for the principal chiral model, parametrized by G/H
x G/H, which correspond to our boundary conditions.
