  Conditional and Lie symmetries of semi-linear 1D Schr\"odinger and diffusion
equations are studied if the mass (or the diffusion constant) is considered as
an additional variable. In this way, dynamical symmetries of semi-linear
Schr\"odinger equations become related to the parabolic and almost-parabolic
subalgebras of a three-dimensional conformal Lie algebra conf_3. We consider
non-hermitian representations and also include a dimensionful coupling constant
of the non-linearity. The corresponding representations of the parabolic and
almost-parabolic subalgebras of conf_3 are classified and the complete list of
conditionally invariant semi-linear Schr\"odinger equations is obtained.
Possible applications to the dynamical scaling behaviour of phase-ordering
kinetics are discussed.
