  The $\eta \to \pi^+ \pi^- \pi^0 \gamma$ decay is discussed in the general
context of Chiral Perturbation Theory (ChPT), assuming that the low--energy
constants (counter--terms) are saturated by vector-meson resonances. The $\eta
\to \pi^+ \pi^- \pi^0 \gamma$ amplitude can be separated in two distinct
pieces: the inner bremsstrahlung, $A^{(IB)}$, and the structure dependent (or
direct emission), $A^{(SD)}$, amplitudes. The former -- which essentially
contains the same physics as $A(\eta \to \pi^+ \pi^- \pi^0)$ -- is found to
dominate over the second one -- which looks more interesting from the ChPT
point of view.
