Efficient construction of contact coordinates for partial prolongations

Let $\text{CV}$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the construction of local coordinates on $M$ such that $\text{CV}$ may be locally expressed as some partial prolongation of the contact distribution $\text{Cal}{C}_q^{(1)}$, on the first order jet bundle of maps from $\Bbb R$to$\Bbb R^q$,$q\geq 1$\mathrm{.}Itisproventhatif$\CV$ is locally equivalent to a partial prolongation of $\text{Cal}{C}_q^{(1)}$ then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on $M$. The number of these first integrals that must be computed satisfies a natural minimality criterion. These results therefore provide a full and constructive generalisation of the classical Goursat normal form from the theory of exterior differential systems.