  We study the graphite intercalated compound CaC$_6$ by means of Eliashberg
theory, focusing on the anisotropy properties. An analysis of the
electron-phonon coupling is performed, and we define a minimal 6-band
anisotropy structure. Comparing with Superconducting Density Functional Theory
(SCDFT) the condition under which Eliashberg theory is able to reproduce the
SCDFT gap structure is determined, and we discuss the role of Coulomb
interactions. The Engelsberg-Schrieffer polaron structure is computed by
solving the Eliashberg equation on the Matsubara axis and analytically
continuing it to the full complex plane. This reveals the polaronic
quasiparticle bands anisotropic features as well as the interplay with
superconductivity.
