Non-linear Schrödinger Equations, Separation and Symmetry

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [1], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers. Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries. For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number. A Lie-algebra approach for non-linear theories is developed.