A constructive generalised Goursat normal form

We provide necessary and sufficient conditions on the derived type of a vector field distribution $\text{Cal}V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\text{Cal}{C}_q^{(1)}$, on the first order jet bundle of maps from $\mathbb{R}$ to ${\mathbb{R}}^q$, $q\ge 1$. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is proven that if $\text{Cal}V$ is locally equivalent to a partial prolongation of $\text{Cal}{C}_q^{(1)}$ then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on the ambient manifold.