  Previous representations of pion pair production amplitudes by two real
photons at low energy, which combine dispersion theoretical constraints with
elastic unitariy, chiral symmetry and soft photon constraints are generalized
to the case where one photon is virtual. The constructed amplitudes display
explicitly the dependence on the $\pi\pi$ phase-shifts, on pion form factors
and on pion polarizabilities. They apply both for space-like and time-like
virtualities despite the apparent overlap of the left and right-hand cuts, by
implementing a definition of resonance exchange amplitudes complying with
analyticity and consistent limiting prescriptions for the energy variables.
Applications are made to the pion generalized polarizabilies, to vector meson
radiative decays, and to the $\sigma\gamma$ electromagnetic form factor.
Finally, and evaluation of the contribution of $\gamma\pi\pi$ states in the
hadronic vacuum polarization to the muon $g-2$ is given, which should be less
model dependent than previous estimates.
