Necessary conditions of the integrability

The definition of an integrable two-dimensional partial differential system is proposed. It contains the two requirements: (i) the triadic representation must exist and (ii) the infinite set of local or non-local conservation laws must arise from the associated linear system. It is proved that each integrable system admits the so-called canonical conserved densities which are expressed in terms of the non-linear system itself. The generalization of the Chen-Lee-Liu algorithm (1979) for calculation of the canonical densities is presented. Several technical results and examples are also considered.