  In this paper we show existence and uniqueness of the solution in viscosity
sense for a system of nonlinear $m$ variational integral-partial differential
equations 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 noise
is driven by a L\'evy process. The switching costs depend on $(t,x)$. As a
by-product of the main result we obtain that the value function of the
switching problem is continuous and unique solution of its associated
Hamilton-Jacobi-Bellman system of equations. The main tool we used is the
notion of systems of reflected BSDEs with oblique reflection driven by a L\'evy
process.
