The constraint on potential and decomposition for (2+1)-dimensional integrable systems

Besides a sufficient condition, a necessary condition is given to determine some kind of consistent constraint on the potential of a (2+1)-dimensional integrable system, which cannot be obtained from the sufficient condition. Under this kind of constraint on the potential a (2+1)-dimensional integrable system can be decomposed into two commuting (1+1)-dimensional integrable systems, and can be further decomposed into three commuting finite-dimensional integrable Hamiltonian systems. The BKP hierarchy is taken as an illustrative example.