We consider an autonomous system of ordinary differential equations, which is resolved with respect to derivatives. To study local integrability of the system near a degenerate stationary point, we use an approach based on Power Geometry and on the computation of the resonant normal form. For the partial non Hamilton 5-parameter case of concrete planar system, we found the complete set of necessary conditions on parameters of the system for which the system is locally integrable near a degenerate stationary point. These sets of parameters, satisfying the conditions, consist of 4 two-parameter subsets in this 5-parameter space. For these subsets we found global first integrals of the system. For the forth subset, at the moment we have not proved analyticity of this first integral.