The eigenmodes in a two-dimensional dimer made of gain and lossy nanoparticles have been investigated within an exact analytical approach. It has been shown that there are eigenmodes for which all Joule losses are exactly compensated by the gain. Among such solutions there are solutions with a new type of symmetry, which we refer to as loss compensation symmetry, as well as well-known parity-time (PT) symmetric solutions. Unlike PT symmetric ones, the modes with loss compensation symmetry allow one to achieve full loss compensation with significantly less gain that in the case of PT symmetry. This effect paves the way to new loss compensation methods in optics.