Some Anosov actions which are affine

Following the works of Y. Benoist, P. Foulon and F. Labourie \cite{BFL}, and having in mind the standing conjecture about the algebricity of Anosov actions of $\mathbb{R}^k$, we propose some geometrical conditions which generalize the notion of contact structures and prove that Anosov actions associated with such structures are conjugated to an Affine action. We also construct two families of examples, the first one is algebraic in nature, and the second one imposes some dynamical conditions instead of geometrical ones.