  Although the Lepton Flavor Violating (LFV) decay $\mu^+\to e^+ \gamma$ is
forbidden in the Standard Model (SM), it can take place within various theories
beyond the SM. If the branching ratio of this decay saturates its present bound
[{\it i.e.,} Br$(\mu^+ \to e^+\gamma)\sim 10^{-11}$], the forthcoming
experiments can measure the branching ratio with high precision and
consequently yield information on the sources of LFV. In this letter, we show
that for polarized $\mu^+$, by studying the angular distribution of the
transversely polarized positron and linearly polarized photon we can derive
information on the CP-violating sources beyond those in the SM. We also study
the angular distribution of the final particles in the decay $\mu^+\to e^+_1
e^- e^+_2$ where $e^+_1$ is defined to be the more energetic positron. We show
that transversely polarized $e_1^+$ can provide information on a certain
combination of the CP-violating phases of the underlying theory which would be
lost by averaging over the spin of $e^+_1$.
