On First Order Congruences of Lines in ${\mathbb{P}}^4$ with Generically Non-reduced Fundamental Surface

In this article we obtain a complete description of the congruences of lines in ${\text{p}}^4$ of order one provided that the fundamental surface $F$ is non-reduced (and possibly reducible) at one of its generic points, and their classification under the hypothesis that (F)_{\red} is smooth.