  The essentially non-perturbative polarization effects are considered for a
planar supercritical Dirac-Coulomb system with strong coupling (similar to
graphene and graphene-based heterostructures) in terms of induced charge
density $\rho_{VP}(\vec{r})$. The main attention is paid to the
renormalization, convergence of the partial expansion and the behavior of
$\rho_{VP}(\vec{r})$ and the integral induced charge $Q_{VP}$ in the
overcritical region. The dependence of the induced density on the screening of
the Coulomb asymptotics of the external source is also explored in detail. Some
peculiar effects in the discrete spectrum with the lowest rotational numbers
$m_j=\pm 1/2\, , \pm3/2$ in the screened case are detected and their possible
role in the transition through corresponding $Z_{cr}$ is also discussed.
