  Conditional 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. 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. Applications to the
phase-ordering kinetics of simple magnets and to simple particle-reaction
models are briefly discussed.
