On symmetry classification of integrable evolution equations of third order

We present new results in the framework of symmetry classification of third order integrable evolution vector equations. A technique proposed by G.A. Meshkov and V.V. Sokolov allowed us to find 12 equations satisfying necessary integrability conditions. We provide a short review of all known equations of the considered type and also clarify all computational difficulties not allowing us to complete the classification problem in the general form.
By imposing reasonable additional restrictions for the form of equations while classifying them we succeed to complete the calculations. The found equations possess several nontrivial conserved densities and they are likely exactly integrable. As the proof of their integrability, the Lax representation or Bäcklund autotransform could serve but finding them is a rather complicated problem requiring a sufficient motivation, for instance, an application value of some of these equations.