  For the flavor-singlet heavy quark system of charmonia in the pseudoscalar
($\eta_c(1S)$) channel, we calculate the elastic (EFF) and transition form
factors (TFF) ($\eta_c(1S) \rightarrow \gamma \gamma^*$) for a wide range of
photon momentum transfer squared ($Q^2$). The framework for this analysis is
provided by a symmetry-preserving Schwinger-Dyson equation (SDE) and
Bethe-Salpeter equation (BSE) treatment of a vector$\times$vector contact
interaction (CI). We also employ an algebraic model (AM), developed earlier to
describe the light quark systems. It correctly correlates infrared and
ultraviolet dynamics of quantum chromodynamics (QCD).
  The CI results agree with the lattice data for low $Q^2$. For $Q^2 \geqslant
Q_0^2$, the results start deviating from the lattice results by more than $20
\%$. $Q_0^2 \thickapprox 2.5 {\rm GeV}^2$ for the EFF and $\thickapprox 25 {\rm
GeV}^2$ for the TFF. We also present the results for the EFF, TFF as well as
$\eta_c(1S)$ parton distribution amplitude for the AM. Wherever the comparison
is possible, these results are in excellent agreement with the lattice,
perturbative QCD, the results obtained through an SDE-BSE study, employing
refined truncations, as well as the experimental findings of the BABAR
experiment.
