Algebraically simple involutive differential systems and Cauchy problem

Systems of polynomial-nonlinear partial differential equations (PDEs) possessing certain properties are considered. Such systems studied by American mathematician Thomas in the 30th of the XX-th century and called him (algebraically) simple. Thomas gave a constructive procedure to split an arbitrary system of PDEs into a finite number of simple susbsystems. The class of simple involutive systems of PDEs includes the normal or Kovalewskaya-type systems and Riquier's orthonomic passive systems. This class admits well-posing of the Cauchy problem. We discuss the basic features of the splitting algorithm, completion of simple systems to involution and posing the Cauchy problem. Two illustrative examples are given. Bibl. – 17 titles.