  In this paper, the completed investigation of a possible superconducting
phase in monolayer indium selenide is determined using first-principles
calculations for both the hole and electron doping systems. The hole-doped
dependence of the Fermi surface is exclusively fundamental for monolayer InSe.
It leads to the extensive modification of the Fermi surface from six separated
pockets to two pockets by increasing the hole densities. For low hole doping
levels of the system, below the Lifshitz transition point, superconductive
critical temperatures $T_c \sim 55-75$ K are obtained within anisotropic
Eliashberg theory depending on varying amounts of the Coulomb potential from
0.2 to 0.1. However, for some hole doping above the Lifshitz transition point,
the combination of the temperature dependence of the bare susceptibility and
the strong electron-phonon interaction gives rise to a charge density wave that
emerged at a temperature far above the corresponding $T_c$. Having included
non-adiabatic effects, we could carefully analyze conditions for which either a
superconductive or charge density wave phase occurs in the system. In addition,
monolayer InSe becomes dynamically stable by including non-adiabatic effects
for different carrier concentrations at room temperature.
