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Propagation of arbitrary-amplitude ion-acoustic solitary waves (IASWs) as well as ion-acoustic periodic waves (IAPWs) is investigated in a fully degenerate quantum electron-ion plasma consisting of isothermal- or adiabatic-ion species. It is shown that the system dimensionality and degrees of freedom play critical roles in matching the criteria for the propagation of such waves. Furthermore, it is revealed that, for the case of the adiabatic-ion species (unlike the isothermal one), in some cases, there exists an upper fractional ion-temperature limit for the existence of IAPWs. It is also shown that the variations of the wave amplitude with respect to the change in the fractional ion temperature are quite different for the cases of the isothermal and adiabatic-ion plasmas.