Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{ sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schr\"{ o}dinger equation and an isotropic SU(3) Landau-Lifshitz equation.
Categories: nlin.SI math-ph math.MP