Global solutions of formally integrable and completely integrable systems of partial differential equations are characterized by means of integral bordism groups. Unfortunately, a thorough acquintance with the author's articles, namely with \textit{A. Prástaro} [Acta Appl. Math. 51, 243--302 (1998; Zbl 0924.58103) and Acta Appl. Math. 59, 111--201 (1999; Zbl 0949.35011)] is necessary. Some fundamental concepts are recalled: the Cartan's approach to differential \(n\)th order equations \(E_k\) which are realized as a subbundle of the \(k\)-jet bundle \(JD^kW\) of a fiber bundle \(\pi:W\to M\) \((\dim M=n)\), then the crucial concept of a singular solution \(V\subset E_k\) which is projected diffeomorphically on \(M\) except a nonwhere dense subset \(\sum(V)\subset V\) of singularities, moreover the \(p\)-dimensional integral manifolds \(N\subset E_k\) that are contained in appropriate solution \(V\), and finally the integral bordism of two \((n-1)\)-dimensional admissible integral manifolds \(N_1,N_2\) defined by \(\partial V=N_1\dot\cup N_2\) together with (a more involved) concept of quantum bordism. The author relates the integral bordism and the quantum bordism groups, by emphasizing the role of singular solutions. Moreover the existence of smooth bording manifolds \(V\) is reduced to the equality of certain characteristic numbers of \(N_1\) and \(N_2\) which are defined in terms of the conservation laws. The Laplace equation \(u_{xx}+u_{yy} u_{zz}=f(x,y,z)\), the Tricomi equation \(u_{yy}=yu_{xx}\) and the d'Alembert equation \(\partial^n\log f/\partial x_1\dots\partial x_n=0\) are discussed.