The characterization of self-adjoint domains for symmetric differential operators with interior singular points in the direct sum spaces was investigated. By constructing different quotient spaces, using the method of symplectic geometry, the self-adjoint extensions of symmetric differential operators in the direct sum spaces for the different deficiency indices at (2,2)singular points was discussed . The classification and description of complete Lagrangian submanifold that correspond with self-adjoint domains of second order differential operators were given. 