  We prove that standing-waves solutions to the non-linear Schr\"odinger
equation in dimension one whose profiles can be obtained as minima of the
energy over the mass, are orbitally stable and non-degenerate, provided the
non-linear term $ G $ satisfies a Euler differential inequality. When the
non-linear term $ G $ is a combined pure power-type, then there is only one
positive, symmetric minimum of prescribed mass.
