  We present a study of photon-photon scattering in the mass range
$W_{\gamma\gamma} < 5$ GeV. We extend earlier calculations of this cross
section for $W_{\gamma\gamma}>$ 5 GeV into the low mass range where
photoproduction of the pseudoscalar resonances $\eta$, $\eta^{'}(958)$
contributes to two-photon final states. We present the elementary photon-photon
cross section as a function of diphoton mass $M_{\gamma\gamma}$ arising from
lepton and quark loop diagrams, and the visible cross section obtained with the
gamma-gamma decay branching fractions of the resonances $\eta$,
$\eta^{'}(958)$, $\eta_c(1S)$, $\eta_c(2S)$, $\chi_{c0}(1P)$. We derive the
corresponding cross sections in ultra-peripheral Pb-Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV by folding the elementary cross section with the
heavy-ion photon fluxes. We consider the dominating background of the
two-photon final state which arises from gamma decays of photoproduced
$\pi^{0}$-pairs. Such $\pi^{0}$-pairs contribute to the background when only
two of the four decay photons are within the experimental acceptance, while the
other two photons escape undetected. We reduce this background by applying cuts
on asymmetries of transverse momenta of the two photons and indicate how the
background can be further suppressed using a multivariate sideband analysis. We
present the cross section for the signal and the background at midrapidity
$|\eta| <$ 0.9, and in the forward rapidity range 2.0 $< \eta <$ 4.5.
