Translation foliations of codimension one on compact affine manifolds

Consider two foliations ${\displaystyle {\left\{calF\right\}}_1}$ and ${\displaystyle {\left\{calF\right\}}_2}$, of dimension one and codimension one respectively, on a compact connected affine manifold ${\displaystyle (M,\nabla )}$. Suppose that ${\displaystyle {\nabla }_{T{\left\{calF\right\}}_1}T{\left\{calF\right\}}_2\subset T{\left\{calF\right\}}_2}$; ${\displaystyle {\nabla }_{T{\left\{calF\right\}}_2}T{\left\{calF\right\}}_1\subset T{\left\{calF\right\}}_1}$ and ${\displaystyle TM=T{\left\{calF\right\}}_1\oplus T{\left\{calF\right\}}_2}$. In this paper we show that either ${\displaystyle {\left\{calF\right\}}_2}$ is given by a fibration over ${\displaystyle S^1}$, and then ${\displaystyle {\left\{calF\right\}}_1}$ has a great degree of freedom, or the trace of ${\displaystyle {\left\{calF\right\}}_1}$ is given by a few number of types of curves which are completely described. Moreover we prove that ${\displaystyle {\left\{calF\right\}}_2}$ has a transverse affine structure.