Non-integrability in non-relativistic theories

Generic non-relativistic theories giving rise to non-integrable string solutions are classified. Our analysis boils down to a simple algebraic condition for the scaling parameters of the metric. Particular cases are the Lifshitz and the anisotropic Lifshitz spacetimes, for which we find that for trivial dilaton dependence the only integrable physical theory is that for z = 1. For the hyperscaling violation theories we conclude that the vast majority of theories are non-integrable, while only for a small class of physical theories, where the Fermi surfaces belong to, integrability is not excluded. Schrödinger theories are also analyzed and a necessary condition for non-integrability is found. Our analysis is also applied to cases where the exponential of the dilaton is a monomial of the holographic coordinate.