This paper contains mainly the results of the diploma thesis of my co-author (written under my supervision). It describes how the algebraic theory of involutive bases may be married with the geometric completion of differential equations according to the Cartan-Kuranishi theorem. By introducing a clever book-keeping, the simple completion algorithm of the Janet-Riquier theory is modified such that it returns intrinsic geometric information. The new algorithm returns precisely the same results as the one underlying our earlier works described in [60, 103] but needs much less computations. The algorithm has been implemented in our MuPAD environment for differential equations (see [105] ).

We present in addition a new and very cheap method to detect singular coordinate systems in which the completion does not terminate. This new method even shows how one may algorithmically construct a regular coordinate system.