Before revision:  A necessary and sufficient condition for integration of order d=1,2,\dots is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0, , d, each one associated with components of the process integrated of order h.
After revision:  A necessary and sufficient condition for integration of order d=1,2,\dots is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0, . An existence theorem shows that the solution of an AR with a unit root of finite type is necessarily integrated of some finite integer d and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the Triangular representation of the process integrated of order h.