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In this paper we show existence and uniqueness of the solution in viscositysense for a system of nonlinear $m$ variational integral-partial differentialequations with interconnected obstacles whose coefficients $(f_i)_{i=1,\cdots,m}$ depend on $(u_j)_{j=1,\cdots,m}$. From the probabilistic point of view,this system is related to optimal stochastic switching problem when the noiseis driven by a L\'evy process. The switching costs depend on $(t,x)$. As aby-product of the main result we obtain that the value function of theswitching problem is continuous and unique solution of its associatedHamilton-Jacobi-Bellman system of equations. The main tool we used is thenotion of systems of reflected BSDEs with oblique reflection driven by a L\'evyprocess.

Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem Driven by L\'evy Process