Integral representation of solution manifolds for over determined systems with one singular and one weak singular lines

In this paper, a over determined system of second-order partial differential equations with one singular and one weak singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular and one weak singular line. Under the condition of compatibility, introducing a new function, we come to a over determined system of partial differential equations of the second order with one singular and one weak singular line of a simpler form. The integral representation of the manifold of solutions of the redefined second-order partial differential system with one singular and one weak singular line is found explicitly through three arbitrary constants, for which initial data problems (Cauchy type problems) can be posed.