  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.
