We present an $\operatorname{SO}(2)$-structure and the associated global exterior differential system existing on the contact Riemannian manifold $\cal S$, which is the total space of the tangent sphere bundle, with the canonical metric, of any given $3$-dimensional oriented Riemannian manifold $M$. This is part of a wider theory which can be studied in any dimension. In this seminar we focus on the first interesting dimension and show several new $\operatorname{SU}(2)$-structures on $\cal S$, following the recent ideas introduced by D. Conti and S. Salamon for the study of $5$-manifolds with special metrics.