Space-time Schr\"odinger symmetries of a post-Galilean particle

We study the space-time symmetries of the actions obtained by expanding the action for a massive free relativistic particle around the Galilean action. We obtain all the point space-time symmetries of the post-Galilean actions by working in canonical space. We also construct an infinite collection of generalized Schr\"odinger algebras parameterized by an integer $M$, with $M=0$ corresponding to the standard Schr\"odinger algebra. We discuss the Schr\"odinger equations associated to these algebras, their solutions and projective phases.