  A partial wave analysis (PWA) of the of the $\eta \pi ^0$ system (where $\eta
\to \gamma \gamma$) produced in the charge exchange reaction $\pi ^-p\to \eta
\pi ^0n$ at an incident momentum of 18 GeV$/c$ is presented as a function of
${\eta \pi ^0}$ invariant mass, $m_{\eta\pi^0}$, and momentum transfer squared,
$t_{\pi^{-}\to\eta\pi}$, from the incident $\pi^-$ to the outgoing ${\eta\pi
^0}$ system. $S$, $P$ and $D$ waves were included in the PWA. The $a_0(980)$
and $a_2(1320)$ states are clearly observed in the overall ${\eta\pi ^0}$
effective mass distribution as well as in the amplitudes associated with $S$
wave and $D$ waves respectively after partial wave decomposition. The observed
distributions in moments (averages of spherical harmonics) were compared to the
results from the PWA and the two are consistent. The distribution in
$t_{\pi^{-}\to\eta\pi}$ for individual $D$ waves associated with natural and
unnatural parity exchange in the $t$-channel are consistent with Regge
phenomenology. Of particular interest in this study is the $P$ wave since this
leads to an exotic $J^{PC}=1^{-+}$ for the $\eta \pi$ system. A $P$ wave is
present in the data, however attempts to describe the mass dependence of the
amplitude and phase motion with respect to the $D$ wave as a Breit-Wigner
resonance are problematic. This has implications regarding the existence of a
reported exotic $J^{PC} = 1^{-+}$ meson decaying into $\eta \pi^0$ with a mass
near 1.4 GeV$/c^2$.
