Abstract: 
We study a unified system of stochastic partial differential equations (SPDEs) with Levy jumps in a forwardbackward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in timevariable and positionparameters over a domain (e.g., a hyperbox or a manifold). A solution to the system is defined by a 4tuple random vectorfield process evolving in time. Since our unified system is a generaldimensional vector one with general nonlinearity and general highorder, the popular computation (e.g., integration by parts) based proof method can not be applied. Thus, we develop an approach to prove the wellposedness of an adapted 4tuple strong solution to the system in a topological space and under a sequence of generalized local linear growth and Lipschitz conditions. As the further investigation of our system, we formulate a nonzerosum stochastic differential game (SDG) problem with general number of players. By a 4tuple solution to the system, we get a Pareto optimal Nash equilibrium policy to the SDG. In
addition, illustrative examples from quantum physics, statistical mechanics, and queueing networks are also presented. 
