  For the family of nonlinear Schr\"odinger equations derived by H.-D.~Doebner
and G.A.~Goldin (J.Phys.A 27, 1771) we calculate the complete set of Lie
symmetries. For various subfamilies we find different finite and infinite
dimensional Lie symmetry algebras. Two of the latter lead to a local
transformation linearizing the particular subfamily. One type of these
transformations leaves the whole family of equations invariant, giving rise to
a gauge classification of the family. The Lie symmetry algebras and their
corresponding subalgebras are finally characterized by gauge invariant
parameters.
