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We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study the local integrability of the system near a degenerate stationary point we use an approach based on the Power Geometry and on the computation of resonant normal forms. For a planar system depending on five parameters we give four sets of conditions on parameters of the system for which it is integrable near the degenerate stationary point and found the corresponding first integrals.