Lepton family number violation is tested by searching for $mu^+to e^+X^0$ decays among the 5.8$times 10^8$ positive muon decay events analyzed by the TWIST collaboration. Limits are set on the production of both massless and massive $X^0$ bosons. The large angular acceptance of this experiment allows limits to be placed on anisotropic $mu^+to e^+X^0$ decays, which can arise from interactions violating both lepton flavor and parity conservation. Branching ratio limits of order $10^{-5}$ are obtained for bosons with masses of 13 - 80 MeV/c$^2$ and with different decay asymmetries. For bosons with masses less than 13 MeV/c$^{2}$ the asymmetry dependence is much stronger and the 90% limit on the branching ratio varies up to $5.8 times 10^{-5}$. This is the first study that explicitly evaluates the limits for anisotropic two body muon decays.
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