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This work is motivated by the rapid development of topological objects implementation in quantum computing technologies and in neural networks design based on Josephson junctions. We have fabricated and studied experimentally dependences of cri tical cur rent Ic of long Josephson junction with topological materials in the weak region on ex ter nal magnetic field, H, under a location of magnetic force microscope (MFM) in a vi cinity of the junction for different spatial distributions of the bias current. Significant deviations of the obtained experimental dependences from the predictions of the k nown theoretical models were found. To get more relevant theoretical description we developed a model which takes in to account existence of inhomogene ous distribution of the critical current along the con tact, the real distribution of the bias current in lead electrodes, and the inhomogeneity of the magnetic field created by the MFM tip. The experimental Ic(H) dependences ob tained at different positio ns of the MFM tip were fitted by the theoretical curves. The set of the model parameters provided the standard deviation between the theoretical and experimental Ic(H) dependencies of less than 0.1% have been found during the fitting. It was found that the Ic(H) shapes strongly depends on the position of the tip which is ex plained by the spatial inhomogeneity of the magnetic field associated with the different magnetization of the tip on its surface. The dynamics of the penetration of Josephson vortices into the long SNS junction under an external homogeneous magnetic field and the field created by the tip of an MFM microscope was investigated within the framework of a high-Q oscillator model. The moments of entry and exit of Josephson vortex were found fro m the results of simulation with the change in the phase shift of oscillations of the MFM tip. Supercon du cting states characterized by different number of vortices inside the junction were de tected. The work was supported by the Russian Science Foundation grant no. 20 69 46013.