The unifying methodologies are based on the construction of `bridges' connecting distinct mathematical theories with each other. The purpose of this paper is to study the relationship between the geo\-me\-tric and algebraic formulation of completely integrable systems of order $k$ and dimension $n$ over a differentiable manifold, in terms of contact $C^{k,n}M$ and co-contact $(C^{k,n}M)^0$ of higher order, as seen in [A. Morimoto, {Prolongation of Geometric Structures}, Math. Inst. Nagoya University, Nagoya, (1969)], to establish an equivalence between both formulations.