  We study all the symmetries of the free Schr\"odinger equation in the
non-commutative plane. These symmetry transformations form an
infinite-dimensional Weyl algebra that appears naturally from a two-dimensional
Heisenberg algebra generated by Galilean boosts and momenta. These infinite
high symmetries could be useful for constructing non-relativistic interacting
higher spin theories. A finite-dimensional subalgebra is given by the
Schr\"odinger algebra which, besides the Galilei generators, contains also the
dilatation and the expansion. We consider the quantization of the symmetry
generators in both the reduced and extended phase spaces, and discuss the
relation between both approaches.
