We ify all the radiant affine $3$-manifolds with virtually solvable holonomy. We ify the radiant affine $3$-manifolds admitting a flat embedding of a surface tangent to the radial flow too. Eventually, we show that if the radiant affine manifold is not one of Benzécri's examples, and if its holonomy group is not virtually solvable, then the periodic orbits of the radial flow are of saddle type. As a corollary, we obtain the ification of the convex radiant affine $3$-manifolds. S. Choi, using our results exposed in the preprint version of this article, obtains the complete ification of closed radiant affine $3$-manifolds, answering a conjecture by Y. Carrière.