ABSTRACT
This chapter examines the definition of the factorization and unitarity equations for quantum S-matrix. The most attractive problems in the investigation of field theories with strong internal symmetries is the possibility of the exact evaluation of their S-matrix and its spectral properties as the result of the existence of large family of classical and quantum conservation laws. Equations describing the condition of factorization of an S-matrix are called themselves factorization equations. Factorization equations are certain homological conditions that describe symplectic structure of completely integrable quantum systems. The chapter shows that with the same quantum or classical factorized S-matrix is associated a large family of different two dimensional completely integrable systems possessing this S-matrix. It presents a list of examples of two dimensional completely integrable system generated by the simplest S-matrices. The chapter explores the structure of the interesting equations called in the continuous case elliptic higher Korteveg de Vries equations.
