  As a matter of fact, the solution space of many systems of ordinary or
partial differential equations in engineering or mathematical physics
"can/cannot" be parametrized by a certain number of arbitrary functions
behaving like potentials. For example, such a parametrization exists for a
control system if and only if it is controllable and may be of high order. The
first set of Maxwell equations admits a first order parametrization by the
4-potential. However, Einstein equations in vacuum do not admit any
parametrization. Finally, the stress equations in continuum mechanics admit a
second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the
case n=2 being the Airy function. The purpose of this paper is to use
unexpected deep results of homological algebra and algebraic analysis in order
to prove for the first time that the stress/couple-stress Cosserat equations
admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions
