Our goal is to find previously unknown binary systems among low-mass dwarfs in the solar neighborhood and to test the search technique. The basic ideas are to reveal the images of stars with significant ellipticities and/or asymmetries compared to the background stars on CCD frames and to subsequently determine the spatial parameters of the binary system and the magnitude difference between its components. For its realization we have developed a method based on an image shapelet decomposition. All of the comparatively faint stars with large proper motions (Vmag>13^m^, {mu}>300mas/yr) for which the 'duplicate source' flag in the Gaia DR1 catalogue is equal to one have been included in the list of objects for our study. As a result, we have selected 702 stars. To verify our results, we have performed additional observations of 65 stars from this list with the Pulkovo 1-m 'Saturn' telescope (2016-2017). We have revealed a total of 138 binary candidates (nine of them from the `Saturn' telescope and SDSS data). Six program stars are known binaries. The images of the primaries of the comparatively wide pairs WDS 14519+5147, WDS 11371+6022, and WDS 15404+2500 are shown to be resolved into components; therefore, we can talk about the detection of triple systems. The angular separation {rho}, position angle, and component magnitude difference {Delta}m have been estimated for almost all of the revealed binary systems. For most stars 1.5''<{rho}<2.5'', while {Delta}m<1.5^m^.