Motor equivalence is a key feature of speech motor control, since speakers must constantly adapt to various phonetic contexts and speaking conditions. The Uncontrolled Manifold (UCM) idea offers a theoretical framework for considering motor equivalence. In this framework coordination among motor control variables is separated into two subspaces, one in which changes in control variables modify the acoustic output and another one in which these changes do not influence the output. Our work develops this concept for speech production using a 2D biomechanical model of the tongue, coupled with a jaw and lip model, for vowel production. We first propose a representation of the linearized UCM based on orthogonal projection matrices. Next we characterize the UCMs of various vocal tract configurations of the 10 French oral vowels using their perturbation responses. We then investigate whether these UCMs describe phonetic classes like phonemes, front/back vowels, rounded/unrounded vowels, or whether they significantly vary across representatives of these different classes. We found they clearly differ between rounded and unrounded vowels, but are quite similar within each category. This suggests that similar motor equivalence strategies can be implemented within each of these classes and that UCMs provide a valid characterization of an equivalence strategy.