Godbillion–Vey classes for a one-dimensional manifold over a local algebra

On a one-dimensional manifold $M$ over local algebra $\mathbb A$ whose canonical foliation is orientable, there exists an $\mathbb A$-valued basis 1-form $\omega$, which satisfies the equation $d\omega=\theta\land\omega$. A real-valued 1-form $\omega$ on a smooth manifold subordinate to the same equation determines a foliation of codimension 1. This makes it possible to define a Godbillon class and a Vey class of $M$ in the same manner as in the foliation theory [9]. In the present paper we find triviality conditions for the Godbillon class and the Vey class of a one-dimensional manifold over $\mathbb A$.