The authors generalize their results presented in their previous papers [1-3] on the system theory study of heat flux in cases of non-identically zero temperature distribution in the heat conducting system at zero moment of time. In addition, the authors generalize Garbai＇s integral equation examined in their paper [4] for the case of non-identically vanishing initial conditions. Similarly, as in the cases of the above mentioned articles, the authors assume that the heat-conducting material characteristics of the substances examined are constant and therefore processes are considered linear.