  Photoproduction of $\pi\eta$-pairs from nucleons has been investigated from
threshold up to incident photon energies of $\approx$~1.4~GeV. The quasi-free
reactions $\gamma p\rightarrow p\pi^0\eta$, $\gamma n\rightarrow n\pi^0\eta$,
$\gamma p\rightarrow n\pi^+\eta$, and $\gamma n\rightarrow p\pi^-\eta$ were for
the first time measured from nucleons bound in the deuteron. The corresponding
reactions from a free-proton target were also studied to investigate
final-state interaction effects (for neutral pions the free-proton results
could be compared to previous measurements; the $\gamma p\rightarrow
n\pi^+\eta$ reaction was measured for the first time). For the $\pi^0\eta$
final state coherent production via the $\gamma d\rightarrow d\pi^0\eta$
reaction was also investigated. The experiments were performed at the tagged
photon beam of the Mainz MAMI accelerator using an almost $4\pi$ coverage
electromagnetic calorimeter composed of the Crystal Ball and TAPS detectors.
The total cross sections for the four different final states obey the relation
$\sigma(p\pi^0\eta)$ $\approx$ $\sigma(n\pi^0\eta)$ $\approx$
$2\sigma(p\pi^-\eta)$ $\approx$ $2\sigma(n\pi^+\eta)$ as expected for a
dominant contribution from a
$\Delta^{\star}\rightarrow\eta\Delta(1232)\rightarrow\pi\eta N$ reaction chain,
which is also supported by the shapes of the invariant-mass distributions of
nucleon-meson and $\pi$-$\eta$ pairs. The experimental results are compared to
the predictions from an isobar reaction model.
