Existence conjecture for complete descriptions of nonholonomic systems

Let $\mathbf{v}$ be the vector field given by formula. Consider the equations and the system, together with the linear constraints

$$\sum_{k=1}^N\alpha_{kj}\dot{x}^j=0,$$

for $k=1,\ldots,M$. Existence conjecture. There exist solutions of the equations such that the system completely describes the behavior of the nonholonomic system subject to these constraints. The conjecture concerns whether the proposed differential system captures the behavior of the corresponding nonholonomic system; the supplied text gives no resolution or further conditions under which the assertion is known.