New boundary conditions for classical integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice, are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank-1 ansatz for the 2*2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be that of dynamical symmetry for the A₁ and BC₂ Calogero-Moser problems. Other physical realizations of our quadratic algebra are also considered.