  The reaction $\gamma p \rightarrow p \pi^0 \eta$ has been studied with the
CBELSA/TAPS detector at the electron stretcher accelerator ELSA in Bonn for
incident photon energies from threshold up to 3.1 GeV. This paper has been
motivated by the recently claimed observation of a narrow structure in the
M$_{N\eta}$ invariant mass distribution at a mass of 1678 MeV/$c^2$. The
existence of this structure cannot be confirmed in the present work. Instead,
for E$_{\gamma}$ = 1400 - 1500 MeV and the cut M$_{p\pi^0} \le 1190 $ MeV/$c^2$
a statistically significant structure in the M$_{p\eta}$ invariant mass
distribution near 1700 MeV/$c^2$ is observed with a width of $\Gamma\approx 35$
MeV/$c^2$ while the mass resolution is $\sigma_{res}$ = 5 MeV/$c^2$. Increasing
the incident photon energy from 1420 to 1540 MeV this structure shifts in mass
from $\approx$ 1700MeV/c$^2$ to $\approx$ 1725 MeV/$c^2$; the width increases
to about 50 MeV/$c^2$ and decreases thereafter. The cross section associated
with this structure reaches a maximum of $\approx$ 100 nb around E$_{\gamma}
\approx$ 1490 MeV (W $\approx $ 1920 MeV), which coincides with the $p a_0$
threshold. Three scenarios are discussed which might be the origin of this
structure in the M$_{p\eta}$ invariant mass distribution. The most likely
interpretation is that it is due to a triangular singularity in the $\gamma p
\rightarrow p a_0 \rightarrow p \pi^0 \eta$ reaction
