  We introduce discrete systems in the form of straight (infinite) and
ring-shaped chains, with two symmetrically placed nonlinear sites. The systems
can be implemented in nonlinear optics (as waveguiding arrays) and BEC (by
means of an optical lattice). A full set of exact analytical solutions for
symmetric, asymmetric, and antisymmetric localized modes is found, and their
stability is investigated in a numerical form. The symmetry-breaking
bifurcation (SBB), through which the asymmetric modes emerge from the symmetric
ones, is found to be of the subcritical type. It is transformed into a
supercritical bifurcation if the nonlinearity is localized in relatively broad
domains around two central sites, and also in the ring of a small size, i.e.,
in effectively nonlocal settings. The family of antisymmetric modes does not
undergo bifurcations, and features both stable and unstable portions. The
evolution of unstable localized modes is investigated by means of direct
simulations. In particular, unstable asymmetric states, which exist in the case
of the subcritical bifurcation, give rise to breathers oscillating between the
nonlinear sites, thus restoring an effective dynamical symmetry between them.
