We establish the existence in the sense of sequences of solutions for a certain system of integro-differential equations in a square in two dimensions with periodic boundary conditions involving the normal diffusion in one direction and the superdiffusion in the other direction in a constrained subspace of \(H^{2}\) for the vector functions via the fixed point technique. The system of elliptic equations contains a second order differential operator, which satisfies the Fredholm property. It is demonstrated that, under certain reasonable technical conditions, the convergence in the appropriate function spaces of the integral kernels implies the existence and convergence in \(H_{c}^{2}(\Omega , {\mathbb R}^{N})\) of the solutions. We generalize our results derived in Efendiev and Vougalter (J Dyn Differ Equ, 2022. https://doi.org/10.1007/s10884-022-10199-2) for an analogous system studied in the whole \({\mathbb R}^{2}\) which involved non-Fredholm operators. Let us emphasize that the study of systems is more complicated than the scalar case.
