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This paper studies in detail different problems concerning the identifiability of the non-stationary version of the MAP2. First, a matrix-based methodology to build equivalent processes is given. Second, a unique, canonical representation of the process, so that the infinite, equivalent versions of a process can be reduced to its canonical counterpart is provided. Finally, a characterization of the process in terms of five descriptors representing moments of the three first inter-event time distributions is given.