A certain aspect of the topological properties of the in plane components of the field of a system of two magnets have been experimentally studied. Vector plots of the inhomogeneous field were obtained using computer processing of magneto-optical images in the longitudinal and transverse sensitivity of the Kerr effect. The topological characteristics of the field were changed by applying a homogeneous bias field in the plane. It is shown that the transition from the initial inhomogeneous configuration to a homogeneous one occurs according to various scenarios. The application of the external field leads to the appearance of new singular points, the number of which is due to Poincaré's theorem on the rotation of the vector field. It is shown that the transition from the initial inhomogeneous configuration to the homogeneous one occurs according to various scenarios through topologically non-equivalent structures and depends on the orientation of the resulting magnetic moment of the system relative to the observation plane.