  In the search for exotic mesons, the GlueX collaboration will soon extract
moments of the $\eta\pi^0$ angular distribution. In the perspective of these
results, we generalize the formalism of moment extraction to the case in which
the two mesons are produced with a linearly polarized beam, and build a model
for the reaction $\vec \gamma p \to \eta \pi^0 p$. The model includes resonant
$S$-, $P$-, $D$-waves in $\eta\pi^0$, produced by natural exchanges. Moments of
the $\eta\pi^0$ angular distribution are computed with and without the
$P$-wave, to illustrate the sensitivity to exotic resonances. Although little
sensitivity to the $P$-wave is found in moments of even angular momentum,
moments of odd angular momentum are proportional to the interference between
the $P$-wave and the dominant $S$- and $D$-waves. We also generalize the
definition of the beam asymmetry for two mesons photoproduction and show that,
when the meson momenta are perpendicular to the reaction plane, the beam
asymmetry enhances the sensitivity to the exotic $P$-wave.
