  Starting from hyperbolic dispersion relations, we present a system of
Roy--Steiner equations for pion Compton scattering that respects analyticity
and unitarity requirements, gauge invariance, as well as crossing symmetry, and
thus all symmetries of the underlying quantum field theory. To suppress the
dependence on the high-energy region, we also consider once- and
twice-subtracted versions of the equations, where the subtraction constants are
identified with dipole and quadrupole pion polarizabilities. We consider the
resolution of the $\gamma\gamma\to\pi\pi$ partial waves by a
Muskhelishvili-Omn\`es representation with finite matching point, and discuss
the consequences for the two-photon coupling of the $\sigma$ resonance as well
as its relation to pion polarizabilities.
