  We study the existence, the stability and the non-degeneracy of normalized
standing-waves solutions to a one dimensional non-linear Schr\"odinger
equation. The non-linearity belongs to a class of algebraic functions
appropriately defined. We can show that for some of these non-linearities one
can observe the existence of degenerate minima, and the multiplicity of
positive, radially symmetric minima having the same mass and the same energy.
We also prove the stability of the ground-state and the stability of normalized
standing-waves whose profile is a minimum of the energy constrained to the
mass.
