  Amplitudes of the form $\gamma^*(q^2)\to \gamma P_1P_2$ appear as
sub-processes in the computation of the muon $g-2$. We test a proposed
theoretical modelling against very precise experimental measurements by the
KLOE collaboration at $q^2=m^2_\phi$. Starting from an exact parameter-free
dispersive representation for the $S$-wave satisfying QCD asymptotic
constraints and Low's soft photon theorem we derive, in an effective theory
spirit, a two-channel Omn\`es integral representation which involves two
subtraction parameters. The discontinuities along the left-hand cuts which, for
timelike virtualities, extend both on the real axis and into the complex plane
are saturated by the contributions from the light vector mesons. In the case of
$P_1P_2=\pi\eta$, we show that a very good fit of the KLOE data can be achieved
with two real parameters, using a $T$-matrix previously determined from
$\gamma\gamma$ scattering data. This indicates a good compatibility between the
two data sets and confirms the validity of the $T$-matrix. The resulting
amplitude is also found to be compatible with the chiral soft pion theorem.
Applications to the $I=1$ scalar form factors and to the $a_0(980)$ resonance
complex pole are presented.
